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.     

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.     

Über die Strukturaufklärung von (Hydroxy-oxo-cyclopentenyl)alkansäuren,den Aldolkondensationsprodukten von Dioxoencarbonsäuren aus Rinderleber

Structure Elucidation of (Hydroxy-oxo-cyclopentenyl)alkanoic Acids, the Aldol-Condensation Products of Dioxoene Acids from Cattle Liver During homogenization of cattle liver the highly instable dioxoene acids 13a , 13b , and 13c are formed. These acids cyclize in alkaline solution to yield pairs of stable (hydroxy-oxo-cyclopentenyl)alkanoic acids, which were isolated as methyl esters 4a / 5a , 4b / 5b , and 4c / 5c . The structures of these compounds were deduced from an enriched 3-mg mixture sample by microchemical reactions combined with a GC/MS analysis of the reaction products. Compound 13a was obtained as methyl ester by oxidation of the methyl ester of the corresponding F-acid with NaOCl. It was not possible to isolate 13a in pure form due to its high sensitivity. Instead of the methyl ester of 13a , 4a and 5a were isolated, of which the structures were established.     

The electronic structure of benzvalene : Photoelectronspectroscopic studies

The photoelectron(PE)spectrum of tricyclo[3.1.0.0^2,6] hex-2-ene(benzvalene 1) has been recorded. The first four bands in the PE spectrum of 1 can be assigned to transitions to ²B₂, ²A₁, ²A₂ and ²B₁ states of 1¹. This assignment is discussed in terms of the results of semiempirical and ab initio calculations on 1. Furthermore the highest occupied MO's of 1 are derived qualitatively from an interaction diagram between a distorted bicyclobutane and an ethylene moiety.     

Acid-base reactions of 1,3,4-thiadiazolines and thiocarbonyl ylides; 1,3,4-thiadiazoline-2-spiro-2′-adamantane

The surprising formation of C₂₂H₃₂N₂S₂ from the title compound 1 at 45°C involves the interaction of the basic adamantanethione S-methylide ($\text{2}$) with its acidic precursor $\text{1}$, in the course of which the anion $\text{6}$ undergoes electrocyclic ring opening; the acid and base functions offer the clue to a prolific chemistry of the thiadiazoline $\text{1}$ and the thiocarbonyl ylide $\text{2}$.     

The preparation of a cabbage reference material for environmental monitoring and food analysis

 Cabbage is frequently used in environmental monitoring and food control, and, hence, cabbage reference materials (RMs) are required for ensuring quality assurance. A cabbage RM was prepared in view of certification of specific elements from the "black list" of high toxicological interest and nutritive importance. All tasks of the RM production (production of the plant material, cutting and freeze-drying, determination of the residual water content, preparation of the RM, homogeneity testing, stability testing, certification measurements) are described in detail. Received: 12 November 1999 / Accepted: 29 January 2000     

Orbital Theory of Heterolytic Fragmentation and Remote Effects on Nitrogen Inversion Equilibria

From a molecular orbital study of model systems we derive the electronic requirements for the Grob fragmentation. The necessary condition for an allowed fragmentation in an X-C₁-C₂-C₃-N system, or the amino cation ⁺C₁-C₂-C₃-N is the level ordering A below S . This in turn is set by maximal through-bond coupling of the empty cation orbital and the nitrogen lone pair. The conformational dependence of through-bond coupling is exactly that derived by Grob, namely parallel orientation of the cation orbital (or the C-X bond), the C₂-C₃-σ-bond, and the N-lone-pair. When the C₁-C₂-C₃ and C₂-C₃-N angles are small, the through-space interaction dominates, reversing the level ordering to S below A , and consequently makes the fragmentation forbidden even though the conformational requirements for it are met. Ring closure becomes allowed. Some examples exploiting this result are presented, as well as procedures for enhancing through-bond coupling and thus fragmentation. The through-bond-effect has also kinetic consequences, allowing the definition of a new type of remote neighbouring group participation operative through bonds and not by direct overlap. The position of equilibria in nitrogen inversion processes should also be influenced by remote substituents which are π-acceptors or donors.