Dirichletsche Reihen auf arithmetischen Progressionen

A characterisation of all real Δ≠0 is given, such that $$\overline {f(\sigma + i\Delta \mathbb{N})} = \overline {f(\sigma + i\mathbb{R})} $$ holds for all Dirichlet seriesf(s)=Σa _n n ^?s and all σ greater than the abscissa of absolute convergence off. For only countably many Δ relation (*) is not valid. By using transcendental results it is shown, that (*) holds in the special case, Δ=1.     

Differing Coordination Modes of (O‐Alkyl)‐p‐Ethoxyphenyldithiophosphonato Ligands in Copper(I), Silver(I) and Gold(I) Triphenylphosphine Complexes

The three (O‐methyl)‐p‐ethoxyphenyldithiophosphonato triphenylphosphine complexes of copper, silver and gold, [(Ph₃P)_nM{S₂P(OMe)C₆H₄OEt‐p}] (M = Cu, n = 2; M = Ag, Au, n = 1) investigated structurally by X‐ray diffraction exhibit remarkable structural differences. The copper compound is a four‐coordinate chelate monomer with Cu–S 2.4417(6) and 2.5048(6) Å; P–Cu–S 104.24(2)–114.01(2)°; Cu–S–P 82.49(3)° and 80.85(2)°. The silver compound is a cyclic dimer with bridging dithiophosphonato ligands and three‐coordinate silver atoms [Ag–S 2.5371(5) and 2.6867(5) Å; P–Ag–S 122.88(2)° and 122.17(2)°; Ag–S–P 89.32(2)° and 103.56(2)°]. The gold compound is monomeric with linear dicoordinate gold [Au–S 2.3218(6) Å; P–Au–S 177.72(2)°, Au–S–P 100.97(3)°].     

Tri- and pentanuclear tungsten-(mu-S)-M clusters (M = W, Cu, Ag)

The thiotungstate [Et4N]2[OW(WS4)2], [Et4N]2.1, containing the linear [[S2W(VI)(mu-S)2]2W(IV)=O] core, was prepared from [Et4N]2[WS4] in the presence of the sulfide scavenger Cd2+. Addition of 1,2-bis(o-diphenylphosphinophenyl)ethane (diphosphine) and Cu+ or Ag+ to solutions of 1 in MeCN/DMF led to coordination of the (diphosphine)Cu/Ag fragments to the terminal sulfido ligands of 1, yielding novel linear pentanuclear, heterometallic clusters [mu-[OW(IV)(DMF)(W(VI)S4)2][M(diphosphine)]2], 2 (M = Cu) and 3 (M = Ag). Along with 2, the trinuclear cluster [[mu-(W(VI)S4)[Cu(diphosphine)(2)]], 4, was also obtained. The molecular and crystal structures of [Et4N]2.1, 2.MeCN, 3.MeCN, and 4.2MeCN.CH2Cl2 have been determined.     

Theory-enforced Re-investigation of the Origin of the Large Metal–Carbon Bond Strength in PdCH2I+ and its reactions with unsaturated hydrocarbons

State-of-the-art ab initio studies demonstrate that the reaction Pd⁺ + CH₃I → PdCH₂I⁺ + H^. is endothermic by ca. 20 kcal/mol, which translates into a bond dissociation energy (BDE) of ca. 83 kcal/mol for the Pd⁺? CH₂I bond. This figure is in agreement with an experimental bracket of 68 kcal/mol < BDE(Pd⁺? CH₂I) < 92 kcal/mol. Based on these findings, the previously studied Pd⁺/CH₃I system was re-investigated, and double-resonance experiments demonstrate that the formation of PdCH₂I⁺ occurs stepwise via PdCH as a reactive intermediate. Further, ion/molecule reactions of PdCH₂I⁺ with unsaturated hydrocarbons are studied, which reveal the formation of carbon–carbon bonds in the gas phase.     

Das Reaktionsverhalten von Cp2NbCl2 gegenüber WF6—Struktur von [Cp2NbCl2]4[WF6]2[WCl6]

The oxidation of Cp₂NbCl₂ with pure WF₆ in SO₂ solution yielded the cationic metallocene species [Cp₂NbCl₂]⁺[WF₆]⁻ essentially in quantitative yield. The same reaction carried out in the presence of either equimolar amounts or a two-fold excess of HCN led to the preparation of the new niobocenium salt [Cp₂NbCl₂]₄⁺[WF₆]²⁻ which was studied by single crystal X-ray diffraction. This compound represents the first example of a structurally characterized metallocene-WF₆⁻ complex, and crystallizes in the tetragonal system: space group, P4₁2₁2(No. 92), a = 11.083(8) Å, c = 48.285 (9) Å; Z = 8; R = 0.0759, R_W = 0.0841. ab]Die Oxidation von Cp₂NbCl₂ mit reinem WF₆ führt in SO₂-Lösung zur Synthese von [Cp₂NbCl₂ ]⁺[WF₆]⁻ in nahezu quantitativer Ausbeute. Die analoge Reaktion führt unter Anwesenheit der äquimolaren Menge oder eines zweifachen Überschusses an HCN zur Ausbildung des Niobocenium-Komplexsalzes [Cp₂NbCl₂]₄⁺ [WF₆]₂⁻[WCl₆]²⁻, von dem eine Röntgenstrukturanalyse angefertigt wurde. Diese Verbindung repräsentiert den ersten structurell charakterisierten Vertreter eines Metallocen-WF₆⁻-Komplexes und kristallisiert im tetragonalen System: Raumgruppe P4₁2₁2 (Nr. 92), a = 11.083(8) Å, c = 48.285(9) Å; Z = 8; R = 0.0759, R_W = 0.0841. kw]Niobium; X-ray diffraction; Oxidation; Metallocenes     

Effect of non-planarity on the spin distribution in the radical anions containing the stilbene or the azobenzene π-system: An ESR and ENDOR Study

The radical anions of 5H-dibenzo[a,d]cycloheptene ( 9 ), 5 H-dibenzo[c,f][1,2]diazepine ( 10 ), 5,6-dihydrodibenzo[a,e]cyclooctene ( 11 ), 5,6-dihydrodibenzo[c,g][1,2]diazocine ( 12 ), and (E)-2,2,5,5-tetramethyl-3,4-diphenylhex-3-ene ( 13 ) were characterized by ESR and ENDOR spectroscopy. Their hyperfine data were compared with those previously reported for radical anions also containing the stilbene or the azobenzene π-system. Whereas the π-spin distribution in the radical anions of the stilbene series is only moderately sensitive to deviations of the π-system from planarity, the radical anions of the azobenzene series respond to steric strain by shifting the π-spin population from the benzene rings to the azo group. This finding is impressively demonstrated by the similar hyperfine data for 9 ^{· ?} and 11 ^{· ?} which contrast with the strongly highly hindered 13 ^{· ?}. A. plausible interpretation is readily provided by the electron affinities of the constituent π-moieties in stilbene and azobenzene. While those of benzene and ethene are both comparatively low, the azo group has a considerably higher electron affinity.     

Uniqueness for Quasi-variational Inequalities

This paper presents a uniqueness result for a quasi-variational inequality QVI(1) that, in contrast to existing results, does not require the projection mapping on a variable closed and convex set to be a contraction. Our basic idea is to find a simple QVI(0), for example a variational inequality, for which we can show the existence of a unique solution. Further, exploiting some nonsingularity condition, we will guarantee the existence of a continuous solution path from the unique solution of QVI(0) to a solution of QVI(1). Finally, we can show that the existence of a second different solution of QVI(1) contradicts the nonsingularity condition. Moreover, we present some matrix-based sufficient conditions for our nonsingularity assumption, and we discuss these assumptions in the context of generalized Nash equilibrium problems with quadratic cost and affine linear constraint functions.