Divisible Effect Algebras

Divisible effect algebras and their relations to convex effect algebras and MV-algebras are studied. A categorical equivalence between divisible effect algebras and rational vector spaces is proved. Infinitesimal, sharp and extremal elements in divisible effect algebras are studied and their relations to properties of the state space are shown.     

Tensor Products of Quantum Structures and Their Applications in Quantum Measurements

Tensor products of quantum logics and effect algebras with some known problems are reviewed. It is noticed that although tensor products of effect algebras having at least one state exist, in the category of complex Hilbert space effect algebras similar problems as with tensor products of projection lattices occur. Nevertheless, if one of the two coupled physical systems is classical, tensor product exists and can be considered as a Boolean power. Some applications of tensor products (in the form of Boolean powers) to quantum measurements are reviewed.     

Lattice animals: Supplementation of perimeter polynomial data by graph-theoretic methods

Application of graph-theoretic methods to new perimeter polynomials for connected clusters on a lattice yields extra data on the total number of clusters and for the coefficients in the series expansion for the mean size of clusters at low densities. The lattices studied are the square, the square with next nearest neighbors, the triangular, and the simple cubic.     

Thrombinwirkung und Fibrinolyse in gereinigten Gerinnungs-systemen

Ohne ZusammenfassungD 21.Die vorstehende Arbeit wurde auf Anregung und unter Leitung von Herrn Professor Dr. Wöhlisch ausgeführt.     

Finitely additive states and completeness of inner product spaces

For any unit vector in an inner product space S, we define a mapping on the system of all -closed subspaces of S, F(S), whose restriction on the system of all splitting subspaces of S, E(S), is always a finitely additive state. We show that S is complete iff at least one such mapping is a finitely additive state on F(S). Moreover, we give a completeness criterion via the existence of a regular finitely additive state on appropriate systems of subspaces. Finally, the result will be generalized to general inner product spaces.     

Synthesis of functionalized 2-alkylidenetetrahydrofurans by cyclization of 1,3-bis(trimethylsilyloxy)-1,3-butadienes with epoxides

The Lewis acid mediated cyclization of epoxides with 1,3-bis(trimethylsilyloxy)-1,3-butadienes, electroneutral equivalents of 1,3-dicarbonyl dianions, results in the formation of 2-alkylidenetetrahydrofurans with a great variety of substitution patterns and functional groups. This includes the synthesis of 2,3'-bifuranylidenes and 7-oxabicyclo[4.3.0]nonanes. The cyclization of dienes with functionalized epoxides containing base-labile groups proceeds with good chemoselectivity. In all reactions, good regio- and E diastereoselectivities are observed. Based on the stereoselectivities observed for reactions of 1,2-disubstituted epoxides, a working hypothesis for the mechanism of the reaction is suggested.     

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.     

Two types of intramolecular addition of an Al-N multiple-bonded monomer LAlNAr' arising from the reaction of LAl with N3Ar' (L = HC[(CMe)(NAr)]2, Ar' = 2,6-Ar2C6H3, Ar = 2,6-iPr2C6H3)

The reaction of beta-diketiminated aluminum(I) monomer LAl with a large bulky azide N3Ar' (L = HC(CMeNAr)2, Ar' = 2,6-Ar2C6H3, Ar = 2,6-iPr2C6H3) in the temperature range from -78 degrees C to room temperature affords two different isomers 2 and 3, which have been characterized by spectroscopic and X-ray structural analyses, as well as elemental analysis. The variable-temperature 1H NMR kinetic studies of this reaction indicate the existence of the monomer LAlNAr' (1) at low temperature and the thermal stability of the compounds increases in the order of 1 < 2 < 3.     

Synthese und Kristallstruktur von [Ba(18-Krone-6)(DMF)4][Cd(Se4)2]

Synthesis and Crystal Structure of [Ba(18-Crown-6)(DMF)₄][Cd(Se₄)₂] The title compound has been prepared by the reaction of a DMF-solution of lithium polyselenide with BaSe₂ and cadmium acetate in the presence of 18-crown-6, forming black crystals. The compound was characterized by IR spectroscopy and by an X-ray structure determination. Space group P2/a, Z = 4, 5392 observed unique reflections, R = 0.048. Lattice dimensions at ?90°C: a = 2021.9(12); b = 1019.8(6); c= 2270.8(14)pm, ß = 106.98(4)°. The structure consists of [Ba(18-crown-6)(DMF)₄]²⁺ ions, in which the barium ions are coordinated by the six oxygen atoms of the crown ether molecule and by four oxygen atoms of the DMF molecules, and of [Cd(Se₄)₂]^2? ions. The cadmium atoms are coordinated by two tetraselenide ions in a chelating fashion.