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).     

Darstellung,Ramanspektren und Kristallstrukturen von V2O3(SO4)2, K[VO(SO4)2] und NH4[VO(SO4)2]

Preparation, Raman Spectra, and Crystal Structures of V₂O₃(SO₄)₂, K[VO(SO₄)₂], and NH₄[VO(SO₄)₂] The oxo-sulfato-vanadates(V) V₂O₃(SO₄)₂, K[VO(SO₄)₂], and NH₄[VO(SO₄)₂] have been prepared as crystals suitable for X-ray structure determination. In all structures sulfate acts as an unidentate ligand only toward a single vanadium atom. The structure of V₂O₃(SO₄)₂ consists of a threedimensional network of pairs of cornershared VO₆ octahedra with one terminal oxygen atom each, and SO₄ tetrahedra. All oxygen atoms of the sulfate ions are coordinated. NH₄[VO(SO₄)₂] and K[VO(SO₄)₂] are isostructural. VO₆ octahedra with one terminal oxygen atom and pairs of sulfate tetrahedra form infinite chains by corner sharing. The chains are weakly interlinked to layers. The sulfate ions are distorted towards planar SO₃ molecules and single oxygen atoms attached to vanadium. This structural detail gives an explanation for the mechanism of the reversible reaction K[VO(SO₄)₂] ? K[VO₂(SO₄)] + SO₃ at 400°C. Raman spectra of the compounds have been recorded and interpreted with respect to their structures. Crystal data: V₂O₃(SO₄)₂, monoclinic, space group P2₁/a, a = 947.2(4), b = 891.3(3), c? 989.1(4) pm, β = 104.56(3)°, Z = 4, 878 unique data, R(R_w) = 0.039(0,033); K[VO(SO₄)₂], orthorhombic, space group P2₁2₁2₁, a = 495.3(2), b = 869.6(9), c = 1 627(1)pm, Z = 4, 642 unique data, R(R_w) = 0,11(0,10); NH₄[VO(SO₄)₂], orthorhombic, space group P2₁2₁2₁, a = 495.3(1), b = 870.0(2), c = 1 676.7(4)pm, Z = 4, 768 unique data, R(R_w) = 0.088(0.083).     

Investigation of the Chemo- and Stereoselectivity of the Ketene-Claisen Rearrangement

A novel type of ketene-Claisen rearrangement in which the precursor of the rearrangement is generated in situ by reaction of optically active allyl thioethers with dichloroketene is described. A characteristic feature of this rearrangement is the excellent chemoselectivity in favor of allyl thioethers vs. allyl ethers, i.e., exclusive chirality transfer of the allylic sulfur moiety is observed with 12, 13 , and 25--27 . The cyclic, optically active allyl thioethers (+)-(R)- 4 and (?)-(S)- 4 and the open-chain allyl thioethers 11--13 rearrange with in situ generated dichloroketene to the optically active thioesters (?)-(S)- 28 , (+)-(R)- 28 , and 31-33 , respectively. A chirality-transfer of > 99% in the cyclic cases (+)-(R)- 4 and (?)-(S)- 4 , and 96--98 % in the open-chain cases 11--13 is observed. Furthermore, the dichloroketene-Claisen rearrangement is characterized by a high asymmetric 1,2-induction. The chiral allylic sulfides 25--27 give the optically active thioesters 36--38 with a 1,2-induction > 99% as determined by NMR-shift experiments.     

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     

Wasserstoffbrücken in 1,1‐Bis(2‐hydroxyethyl)‐3‐benzoylthioharnstoff und seinen Nickel(II)‐ und Kupfer(II)‐Chelat‐Komplexen

Hydrogen Bonds in 1,1‐Bis(2‐hydroxyethyl)‐3‐benzoylthiourea and its Nickel(II)‐ and Copper(II)‐Chelate Complexes The ligand 1,1‐bis(2‐hydroxyethyl)‐3‐benzoylthiourea HL, ( 1 ), yields with nickel(II) and copper(II) ions neutral complexes [NiL₂], ( 2 ), and [CuL₂], ( 3 ). By X‐ray structure analysis and IR spectroscopy different intramolecular hydrogen bonds (OH…O) and (OH…N) could be identified in both equally coordinated ligands of the [NiL₂] molecule. For comparison X‐ray and IR data were also estimated for 1 and 3 .     

Solvothermal Synthesis and Characterization of the New Iron Thioantimonates(III) [Fe(C6H18N4)]FeSbS4 and [Fe(C4H13N3)2]Fe2Sb4S10 Containing FeII and FeIII and Protein‐Analogous [2FeIII‐2S]2+ Clusters

The two new compounds [Fe(tren)]FeSbS₄ ( 1 ) (tren = tris(2‐aminoethyl)amine) and [Fe(dien)₂]Fe₂Sb₄S₁₀ ( 2 ) (dien = diethylendiamine) were prepared under solvothermal conditions and represent the first thioantimonates(III) with iron cations integrated into the anionic network. In both compounds Fe³⁺ is part of a [2Fe^III‐2S] cluster which is often found in ferredoxines. In addition, Fe²⁺ ions are present which are surrounded by the organic ligands. In ( 1 ) the Fe²⁺ ion is also part of the thioantimonate(III) network whereas in ( 2 ) the Fe²⁺ ion is isolated. In both compounds the primary SbS₃ units are interconnected into one‐dimensional chains. The mixed‐valent character of [Fe(tren)]FeSbS₄ was unambiguously determined with Mössbauer spectroscopy. Both compounds exhibit paramagnetic behaviour and for ( 1 ) a deviation from linearity is observed due to a strong zero‐field splitting. Both compounds decompose in one single step.     

EVIDENCE FOR A MAGNESIUM DEPENDENT ATPase IN BOVINE ROD OUTER SEGMENT DISK MEMBRANES*†

Abstract—In the presence of Mg²⁺ and adenosine triphosphate (ATP), a rapid. light-induced, light-scattering transient is observed from bovine rod outer segments (ROS). This light-scattering transient we have labelled 'A'. Ca²⁺ cannot replace Mg²⁺. nor can guanosine triphosphate (GTP) replace ATP. 'A' is observed at ATP concentrations as low as a few μM.
The half-time of 'A', 60 ms at 20° and 20 ms at 37°, is consistent with a process possibly involved in visual transduction.
'A' has the action spectrum of rhodopsin bleaching and its amplitude is strictly proportional to the fraction of rhodopsin bleached per flash. 'A' can be regenerated by 11- cis retinal.
Inhibition studics with ATP analogues, which cannot be hydrolysed and fail to evoke an 'A' response, reveal that an ATP hydrolysis process has to precede illumination in order for 'A' to occur.
On the basis of the above findings. it is proposed that there is a Mg²⁺ dependent ATPase in ROS that allows the disk membrane to assume a new membrane state which, upon illumination, is altered. giving rise to the structural phenomenon monitored as light-scattering transient 'A'.     

Zur Theorie der harmonischen Differentialformen

Generalizing harmonic differential forms (rot =0, div =0 in M, M being a smooth riemannian manifold with boundary) of first and second kind (=0 and *=0 on M resp.) within the framework of Hilbert space notation, it is possible to extend the meaning of the boundary conditions to non-smooth boundaries. It turns out that in this case the classical result is still valid for certain open subregions G of M: The dimension of the space of harmonic differential forms of second kind is given by the q-th Betti number of G; *-duality leads to the respective result for harmonic differential forms of first kind.     

Gewöhnliche lineare Differentialgleichungen zweiter Ordnung mit Distributionskoeffizient

Zusammenfassung Läßt man in der gew. lin. Dgl. zweiter Ordnung einen Distributionskoeffizient zu, der Ableitung einerL _loc ² -Funktion ist, so existieren eindeutig festlegbare, stetige Lösungen derart, daß die in der Dgl. auftretenden Produkte definiert werden können. Mit einer Regularisierung des Distr.-Koeffizienten können die Lösungen durchC -Funktionen approximiert werden. Es wird der Sturmsche Vergleichssatz verallgemeinert. Hieraus und aus den Eigenschaften der mit zwei Basislösungen gebildeten Lösungskurve folgt die Gültigkeit des Ljapunovschen Satzes über die Eigenwertverteilung der verallgemeinerten Hillschen Dgl. mit positivem Distr.-Koeffizient, falls eine zusätzliche Bedingung erfüllt ist. Die vorliegende Arbeit enthält einige Ergebnisse meiner Dissertation [6] (weitere Resultate und Details finden sich dort).     

Windschiefe Kongruenzflächen konstanten Dralls

In this paper we study the curves of a surface ( _T -lines) which are base curves of ruled surfaces for which the parameter of distribution O has a constant value. Moreover we assume that the ruled surfaces belong to a given congruenceT of surface tangents of . Relations are established between the _T -lines and other curves of (asymptotic lines, lines of curvature, k _T -lines). The _T -lines are used to characterize the pseudospheres and the helicoids. The _T -lines of the surfaces of revolution are determined if the congruenceT consists of the tangents of the circles of latitude.

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Herrn WERNER BURAU zum 70.Geburtstag