On the central limit theorem in R k

Summary Let F ^*n denote the n th convolution of a distribution function F on R ^k and suppose that F has zero moments of the first order and finite second order moment matrix. It is well-known that F ^*n (n·) converges to a Gaussian d.f. as n + t8. These d.f.s determine measures F ^*n (nA) and (A) for Borelsets A, We present a method that admits the estimation of the remainder-term F ^*n (n A)- (A) when A belongs to a certain class of Borelsets. This class contains all convex sets. If F has finite absolute third order moments then the remainder-term is of the order n ^–1/2. Also the remainder term's dependence on the dimension k is given. These results strengthen and generalize earlier results in the same direction.This paper was first communicated at the Scandinavian mathematical congress in Oslo, August 1968.     

Beiträge zur Kenntnis des chromophoren Systems der Corrine. X. Über die Reaktion von Derivaten des Cobyrinsäure-heptamethylesters mit unterchloriger Säure

Contributions to the Knowledge of the Corrin Chromophore, X. Reaction of Derivatives of Cobyrinic Acid Heptamethyl Ester with Hypochlorous Acid (Cobalt)-dicyano-, chloro-cyano-, and dichloro-cobyrinic acid heptamethylesters ( 1, 2 and 3 ), respectively, as well as their 10-chloro derivatives 4, 5 and 6 react with hypochlorous acid. A report on the structure and some properties of the reaction products 8, 9 and 10 is given.     

Die Kristallstruktur des Calciumazid-Dihydrates

Single crystals of Ca(N₃)₂·2 H₂O have been prepared from aqueous solutions at room temperature. The crystals are monoclinic,a=1 159.0 (3),b=614.2 (2),c=785.5 (2) pm, =106.52 (2)°,Z=4, space group P2₁/n. The crystal structure was determined by single crystal X-ray diffraction (1 109 Mo-K-reflexions,R=0.052). Calcium atoms are surrounded by four azide groups and four water molecules. The coordination polyhedra are antiprism which are sharing azide groups and water molecules to form layers. The lattice constants and powder pattern agree well with values reported earlier for Ca(N₃)₂ · 1.5 H₂O [1]. It was also shown, that Sr(N₃)₂ · 2 H₂O is isotypic with Ca(N₃)₂ · 2 H₂O.

     

Hoch-Energie Lösungen der Dirac-Gleichung und Berechnung der Regge-Pole

It is shown that the Dirac-equation can be solved for a superposition of Yukawapotentials by a modified perturbation procedure. In particular high-energy asymptotic expansions are obtained for the solutions and the Regge-trajectories. Finally the behaviour of the Regge-poles is discussed.     

Microwave spectrum of methoxyacetic acid: assignment of the hydrogen-bonded rotamer

Microwave spectra of CH₃OCH₂COOH and CH₃OCH₂COOD are reported. One conformation has been assigned. This form of the molecule has a planar HCOCCOOH skeleton with four out-of-plane hydrogens. A weak five-membered intramolecular hydrogen bond is formed between the hydroxyl proton and the ether oxygen thus stabilizing the planar form. Absolute intensity measurements and arguments based on a few reasonable assumptions have been used to show that the assigned rotamer is present at concentrations amounting to between 10 and 30 per cent of the total. Other forms are not identified. Seven vibrationally excited states were assigned and attributed to the three lowest torsional modes. The dipole moment was determined to be μ_a = 4.72±0.04 D, μ_b = 0.15±0.02 D, and μ_total = 4.72±0.04 D.