Laguerre and Minkowski planes produced by dilatations

We show that each parabolic curve f in R² produces a Laguerre plane if f and all its images under dilatations are cycles. Likewise, two hyperbolic curves f₁,f₂ produce a Minkowski planeM(f₁,f₂). We determine for which curves is miquelian resp. ovoidal, and for which pairs f₁,f₂,M(f₁,f₂) is miquelian resp. satisfies the rectangle axiom, thus providing many examples of non-embeddable planes.     

Beiträge zur Untersuchung anorganischer nichtstöchiometrischer Verbindungen. XXIII. Neue Bauelemente bei metastabilen Blockstrukturen im System Nb2O5/WO3 Elektronenmikroskopische Untersuchung in Hochauflösung

Contributions to the Investigation of Inorganic Non-stoichiometric Compounds. XXIII. New Building Elements of Metastable Block Structures in the System Nb₂O₅/WO₃, Investigation by High Resolution Electron Microscopy Under equilibrium conditions the region of existence of block structures in the system Nb₂O₅/WO₃ ends with a phase having the composition 9 Nb₂O₅:8 WO₃ corresponding to values of O/ΣM = 2.654 (M = Nb, W) and W/Nb = 0.44. We now investigated to what extent this region of existence of block structures could be expanded towards a higher content of WO₃. Starting out from the known phases Nb₂O₅:WO₃ = 6:1, 7:3, 8:5 and 9:8 in which Nb and W are present in their maximum states of oxidation, we obtained series of solid solutions by substituting W for Nb. Thereby the initial structure (block sizes [3×4], [4×4], [4×5] and [5×5] M–O-octahedra) and also the respective O/ΣM remained unchanged. Upon complete oxidation of these series of solid solutions which led into reduced systems (e. g. NbO₂/Nb₂O₅/WO₃) at ～500°C metastable products were gained, which also have block structures. Corresponding to the extent of substitution of the initial series of solid solution and the resulting ratio O/ΣM of the oxidation products the new structures are to a wide extent built of large blocks that have an extension (at maximum [5X15] M–O-octahedra) we have not observed so far. These blocks could be seen using high resolution transmission electron microscopy. The investigations on the oxidation products of the solid solutions of thc 8:5 and 9:8 series and illso it model concerning the mechanism of oxidation are described.     

Beiträge zur Untersuchung anorganischer nichtstöchiometrischer Verbindungen. XXII. Neue metastabile Blockstrukturen im System Nb2O5/WO3 Elektronenoptische Untersuchung

Contributions to the Investigation of Inorganic Non-stoichiometric Compounds. XXII. New Metastable Block Structures in the System Nb₂O₅/WO₃, Electron Optical Investigation We succeeded in considerably expanding the region of existence of block structures. By substituting W for Nb, while the ratio O/∑M (M ? Nb, W) is kept constant, and starting from the known phases Nb₂O₅: WO₃ = 6:1, 7:3, 8:5 and 9:8 one obtains series of solid solutions whose metastable products of oxidation have block structures too. In contrast to the solid solutions which have the structures of the starting phases i. e. with socalled blocks of [3 × 4], [4 × 4], [4 × 5] and [5 × 5] M? O-octahedra, the products of the oxidation have structures in which some edge sharing octahedra changed their connections to become octahedra sharing corners thus opening up the possibility for the formation of blocks with twice the number of octahedra as before. Rows of these large blocks with e. g. [4 × 6] or [4 × 8] octahedra alternate nonperiodically with rows of smaller blocks of the initial size. Details of these heavily disordered structures could only be discerned with the help of high resolution electron microscopy.