A remark on almost sure convergence of weighted sums

Summary As a generalization of a theorem of Chow [1] it is shown by an elementary method that for i.i.d. r.v.'s X ₁,...,X _n, with expectation zero and finite p-th absolute moment (p2) the weighted sums converge to zero a.s.     

Cotangent functors of curve singularities

Let R be a complete analytic k-algebra of dimension one, where k is a field of characteristic 0. For a Noetherian normalization , the cotangent functors T_i(R/A, R) are of finite length. It is shown that if R is smoothable, then for all j, where denotes the complementary module. This result is used to give estimates for the length of the torsion of the module of differentials _R/k.     

Synthese und Kristallstrukturen der Phosphanimin-Komplexe MCl2(Me3SiNPMe3)2 mit M = Zn und Co,und CoCl2(HNPMe3)2

Syntheses and Crystal Structures of the Phosphaneimine Complexes MCl₂(Me₃SiNPMe₃)₂ with M = Zn and Co, and CoCl₂(HNPMe₃)₂ The molecular complexes MCl₂(Me₃SiNPMe₃)₂ (M = Zn, Co) have been prepared by the reaction of the dichlorides of zinc and cobalt with Me₃SiNPMe₃ in CH₃CN and CH₂Cl₂, respectively, whereas the complex CoCl₂(HNPMe₃)₂ has been prepared by the reaction of CoCl₂ with NaF in boiling acetonitrile in the presence of Me₃SiNPMe₃. All complexes were characterized by IR spectroscopy and by crystal structure determinations. The complexes MCl₂(Me₃SiNPMe₃)₂ crystallize isotypically. ZnCl₂(Me₃SiNPMe₃)₂: Space group P2₁2₁2₁, Z = 4, 2677 observed unique reflections, R = 0.024. Lattice dimensions at ?70°C: a = 1243.6; b = 1319.0; c = 1464.7 pm. CoCl₂(Me₃SiNPMe₃)₂: Space group P2₁2₁2₁, Z = 4, 3963 observed unique reflections, R = 0,071. Lattice dimensions at ?80°C: a = 1236.3; b = 1317.4; c = 1457.6 pm. CoCl₂(HNPMe₃)₂ · CH₂Cl₂: Space group Pbca, Z = 8, 1354 observed unique reflections, R = 0.055. Lattice dimensions at ?80°C: a = 1247.3; b = 998.4; c = 2882.4 pm. All complexes have monomeric molecular structures, in which the metal atoms are coordinated in a distorted tetrahedral fashion by the two chlorine atoms and by the nitrogen atoms of the phosphaneimine molecules.     

Beryllocene, (C5H5)2Be. The He(I) photoelectron spectrum and ab initio molecular orbital calculations

Ab initio molecular orbital calculations with a double-ζ basis have been carrried out on five models of beryllocene, Cp₂Be, with fixed geometries. The lowest energies are obtained for the π-Cp, σ-Cp and D_5d models.

The He(I) photoelectron spectrum of Cp₂Be was recorded and the ionization potentials of the first bands were compared with the orbital energies obtained from the molecular orbital calculations. A satisfactory fit between experiment and calculations was obtained for a slip sandwich model of C_s symmetry. A model of C_5v symmetry is only compatible with the PE spectrum if the Jahn—Teller splitting of the lowest ²E₁ state of the molecular ion is exceptionally large, 1.0 eV.     

A Green's function approach to the photoelectron spectrum of bis(π-allyl)nickel[1]

The vertical ionization potentials of bis(-allyl)nickel (see (1) in Fig. 1) are calculated by means of the Green's function approach within a semiempirical INDO extension to the first transition metal series. The computed ionization potentials are in good agreement with an experimentally deduced assignment. In contrast to earlier theoretical and experimental studies, the 7a _u () level is predicted on top of the levels corresponding to the Ni 3d orbitals. Our approach leads to a complete assignment of the PE spectrum of (1) in the outer valence region.     

Asymmetric Catalysis by Vitamin B12. The Mechanism of the Cob(I)alamin-Catalyzed Isomerization of 1,2-Epoxycyclopentane to (R)-Cyclopent-2-enol

The isomerization of 1,2-epcxycyclopentane ( 1 ) to enantiomerically enriched (R)-cyclopent-2-enol ( 2 ) in protic solvents is catalyzed by cob(I)alamin. The enantiomeric excess (e.e.) of (R)- 2 is usually ca. 60%; it is only slightly dependent on the temperature, but increases with decreasing dielectric constant ε of the solvent. Standard kinetic methods show the reaction to be first order in vitamin B₁₂ and zero order in 1 . The rate constant increases exponentially with increasing ε of the solvent. An Arrhenius plot at ε = 40 gives activation parameters ΔH^≠ = 78 ± 4 kJ·mol^?1 and ΔS^≠ = ?49 ± 1 J·mol^?1·K^?1. The isomerization 1 → 2 proceeds in two steps (Schemes 2 and 7): (i) The epoxide ring is first opened by the proton-assisted fast and irreversible nucleophilic attack of the chiral Co^I catalyst to form diastereoisomeric (1R,2R)- and (1S,2S)-(2-hydroxycyclo-pentyl)cob(III)alamins 6 in a ratio of ca. 4:1 which are the dominant species in the steady state; (ii) The intermediates 6 then decompose in the rate-limiting step to form 2 and recycled catalyst. Experiments with specifically ²H-labeled 1 showed the hydro-cobalt elimination 6 → 2 to be non-stereoselective. It proceeds via reversible Co? C bond homolysis to a free 2-hydroxycyclopentyl radical from which stereoelectronically controlled H-abstraction by Co¹¹ takes place.