Rechtseiträume

Rectangular spaces are defined to be incidence spaces on which an equivalence relation is defined on the linesB and a congruence relation is defined on the point pairsP ² which is compatible with the relation and the incidence structure. Every rectangular space which contains a line with only a finite number of points is a euclidean space. In any regtangular space of characteristic 2 there exists a unique reflection on each line. Thus, as in the case of rectangular planes, rectangular spaces of arbitrary dimension and of characteristic 2 can be characterized in terms of commutative kinematic spaces with involutory automorphisms.Our main result is the theorem which states that any rectangular space of characteristic 2 can be embedded into a euclidean space of equal dimension. From this embedding property we conclude that every rectangular space of characteristic 2 can be described as a subgeometry of a vector space supplied with a quadratic form. Finally we present examples of rectangular spaces of arbitrary dimension which are not euclidean spaces.

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Herrn Prof. H. Karzel und Herrn Prof. H. Wähling in Dank gewidmet

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Diese Arbeit ist in einer anderen Form in meiner Habilitationsschrift Zur Theorie der Rechtseiträume unter besonderer Berücksichtigung des ebenen Falles (Technische Universität München 1987) enthalten.     

Investigation of the Crystallisation Behaviour of Lead Titanate (PT), Lead Zirconate (PZ) and Lead Zirconate Titanate (PZT) by EXAFS-Spectroscopy and X-Ray Diffraction

Two solid solutions of lead zirconium titanates PbZr _x Ti_{1 – x} O₃ (x = 0.1 and 0.35) as well as the reference compounds lead titanate and lead zirconate were prepared from zirconium and titanium n-propoxide, dissolved in 2-methoxyethanol, by sol-gel process. The amorphous products after pyrolysis of the dried gels and the crystalline phases were studied by EXAFS spectroscopy to monitor the structural changes from the amorphous oxide mixture to the crystalline ceramics after calcination. Additionally, the crystalline phases were identified by X-ray diffraction (XRD).It follows from the analysis of the EXAFS data that the local order of the amorphous phases seems to be completely different from that of the crystalline phase. There is no indication of a preformation of the local order of the perovskite structure. The analysis of our EXAFS spectra can be interpreted very consistently with the assumption that in the amorphous samples a segregation exists on molecular level and the low crystallisation temperatures are a consequence of very short diffusion paths.     

Anordnung in Rechtseitebenen und Integritätssystemen

H. Karzel [6] has shown that rectangular planes can be algebraically represented by metric integral systems. Here, rectangular planes, which are simultaneously ordered planes, are considered. Necessary and sufficient conditions on the integral system are given for the corresponding rectangular plane to be ordered.Moreover, it is shown that the maximal multiplicative system of an ordered rectangular plane is a valuation ring.     

First principles calculations of oxygen reduction reaction at fuel cell cathodes

The efficiency of solid oxide fuel cells (SOFC) depends critically on materials, in particular for the cathode where the oxygen reduction reaction (ORR) occurs. Typically, mixed conducting perovskite ABO₃-type materials are used for this purpose. The dominating surface terminations are (001) AO and BO₂, with the relative fractions depending on materials composition and ambient conditions.Here, results of recent large-scale first principles (ab initio) calculations for the two alternative polar (La,Sr)O and MnO₂ (001) terminations of (La,Sr)MnO₃ cathode materials are discussed. The surface oxygen vacancy concentration for the (La,Sr)O termination is more than 5 orders of magnitude smaller compared to MnO₂, which leads to drastically decreased estimated ORR rates. Thus, it is predicted for prototypical SOFC cathode materials that the BO₂ termination largely determines the ORR kinetics, although with Sr surface segregation (long-term degradation) its fraction of the total surface area decreases, which slows down cathode kinetics.     

How Is Oxygen Incorporated into Oxides? A Comprehensive Kinetic Study of a Simple Solid‐State Reaction with SrTiO3 as a Model Material

The kinetics of stoichiometry change of an oxide--a prototype of a simple solid-state reaction and a process of substantial technological relevance--is studied and analyzed in great detail. Oxygen incorporation into strontium titanate was chosen as a model process. The complete reaction can be phenomenologically and mechanistically understood beginning with the surface reaction and ending with the transport in the perovskite. Key elements are a detailed knowledge of the defect chemistry of the perovskite as well as the application of a variety of experimental and theoretical tools, many of them evolving from this study. The importance of the reaction and transport steps for (electro)chemical applications is emphasized.     

On the proton conductivity in pure and gadolinium doped nanocrystalline cerium oxide

Conductivity measurements were performed on microcrystalline and nanocrystalline ceria (undoped and doped) in dry as well as wet atmosphere. Below 200-250 °C, the nanocrystalline samples exhibit an enhanced total conductivity under wet conditions, which increases with decreasing temperature. In addition, thermo-gravimetric analysis revealed a strong water uptake below 200 °C. DC-polarization measurements confirm the ionic character of conductivity in the nanocrystalline samples at low temperatures. The role of both grain boundaries and residual porosity on the enhanced conductivity below 200 °C is discussed.     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

Eine Kennzeichnung der nicht fanoschen Rechtseitebenen durch Inzidenz,Parallelität und Kongruenz

Rectangular planes of characteristic 2 in the sense of H. KARZEL [7] will be characterized as incidence spaces with parallelism and congruence .