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.     

The distribution of nickel ions among octahedral and tetrahedral sites in NiAl2O4MgAl2O4 solid solutions

The cation distribution in NiAl₂O₄ and in the solid solutions Ni_xMg_1?xAl₂O₄ with x ranging from 0.01 to 0.55 has been studied by X-ray analysis, magnetic susceptibility and reflectance spectroscopy.The relative X-ray intensities of various reflections for each compound were measured and compared with intensities calculated for various models of cation distribution. Lattice parameters, magnetic moments of Ni²⁺, and the position of nickel absorption bands have also been measured. The results show that all specimens have a predominantly octahedral distribution for their nickel ions, and that there is a small change towards a random distribution when the quenching temperature is increased from 1273 to 1673°K.Both X-ray and magnetic results show that the fraction, α, of Ni²⁺ ions on tetrahedral sites varies with nickel content. For samples quenched from 1273°K, NiAl₂O₄ is 78% inverse, but between x = 1.0 and x = 0.25 the value of α decreases from 0.22 towards 0.1. Thus, the fraction of nickel on octahedral sites in the solid solutions is higher than in NiAl₂O₄. The effect is linked with an increase in Dq for the Ni²⁺ ion. Below x = 0.25, there is tentative evidence that α may increase. In nickel-dilute solutions the matrix as a whole is nearly normal, and it is suggested that Madelung energy may then be more effective in counteracting the octahedral stabilization of nickel. The same trend is observed in the specimens quenched from 1673°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.