Jung's Theorem for Alexandrov Spaces of Curvature Bounded Above

The classical Jung theorem gives an optimal upper estimate for the radius of a bounded subset of R ⁿ in terms of its diameter and the dimension. In this note we present an analogue of this result for metric spaces of curvature bounded above in the sense of Alexandrov.     

Riemann-Roch-Transformationen und Kobordismen

Ohne Zusammenfassung     

Analytical results for mixed action SU(2) lattice gauge theory

Strong and weak coupling expansions for the SU(2), mixed action lattice gauge theories are given. Nonperturbative configurations are identified and their contribution is also estimated. A comparison with Monte Carlo data is presented.     

Vergleich der Produkte aus der Reaktion von Phenylguanidin-Derivaten mit β-Ketoestern bzw. Propiolsäureestern (Synthese von Pyrimidonen)

Comparison of reaction products afforded by phenylguanidine derivatives with β-ketoesters or propiolic esters, respectively (synthesis of pyrimidones). 2-Anilino-imidazolines, when treated with either β-ketoesters or propiolic esters, yield two isomeric groups of pyrimidones. The isomerism is based on different positions of the carbonyl group in the pyrimidone ring. The mass spectra permit an unequivocal assignment of constitution I to the product formed with β-ketoesters and of constitution V to that formed with propiolic esters. Additionally, 2-(2-amino-anilino)-2-imidazoline (XII) when treated with methyl phenylpropiolate yields IX; while treatment with ethyl benzoylacetate yields XIII as an intermediate, which eliminates spontaneously one molecule of water to give the benzimidazole derivative XIV. Phenylguanidines (XV) add propiolic esters in the same way as do imidazoline derivatives. Photochemical cleavage of two carbon atoms with their adherent hydrogen atoms from the imidazole ring of the pyrimidones (V) leads to aminopyrimidine derivatives, e.g. XVI.     

The formation of a new mixed cobalt rhodium carbonyl from Co2(CO)8 and Rh4(CO)12: Infrared spectroscopic characterization under carbon monoxide pressure

The formation of a new compound, the most characteristic IR absorption bands of which appear at 2007 cm^-1 and 1956 cm^-1, has been in the reaction between Co₂(CO)₈ and Rh₄(CO)₁₂ under carbon monoxide pressure in a hydrocarbon medium. The same compound is also formed either by the reaction of Co₂(CO)₈ with [Rh(CO)₂Cl]₂ or by the reaction of Co₃Rh(CO)₁₂ with carbon monoxide. The new complex has not been isolated in a pure state, but the formula CoRh(CO)₇ is proposed on the basis of the stoichiometry of its formation and its physico-chemical properties. Equilibrium constants and thermo-dynamic parameters for the reaction 2 Co₂(CO)₈ + Rh₄(CO)₁₂  4 CoRh(CO)₇ have been estimated. Possible structures for the new complex are discussed on the basis of its IR spectrum.     

One‐dimensional hydrogen‐bonded structures in the 1:1 proton‐transfer compounds of 4,5‐dichlorophthalic acid with 8‐hydroxyquinoline, 8‐aminoquinoline and quinoline‐2‐carboxylic acid (quinaldic acid)

The structures of the 1:1 proton‐transfer compounds of 4,5‐dichlorophthalic acid with 8‐hydroxyquinoline, 8‐aminoquinoline and quinoline‐2‐carboxylic acid (quinaldic acid), namely anhydrous 8‐hydroxyquinolinium 2‐carboxy‐4,5‐dichlorobenzoate, C₉H₈NO⁺·C₈H₃Cl₂O₄⁻, (I), 8‐aminoquinolinium 2‐carboxy‐4,5‐dichlorobenzoate, C₉H₉N₂⁺·C₈H₃Cl₂O₄⁻, (II), and the adduct hydrate 2‐carboxyquinolinium 2‐carboxy‐4,5‐dichlorobenzoate quinolinium‐2‐carboxylate monohydrate, C₁₀H₈NO₂⁺·C₈H₃Cl₂O₄⁻·C₁₀H₇NO₂·H₂O, (III), have been determined at 130 K. Compounds (I) and (II) are isomorphous and all three compounds have one‐dimensional hydrogen‐bonded chain structures, formed in (I) through O—H...O_carboxyl extensions and in (II) through N⁺—H...O_carboxyl extensions of cation–anion pairs. In (III), a hydrogen‐bonded cyclic R₂²(10) pseudo‐dimer unit comprising a protonated quinaldic acid cation and a zwitterionic quinaldic acid adduct molecule is found and is propagated through carboxylic acid O—H...O_carboxyl and water O—H...O_carboxyl interactions. In both (I) and (II), there are also cation–anion aromatic ring π–π associations. This work further illustrates the utility of both hydrogen phthalate anions and interactive‐group‐substituted quinoline cations in the formation of low‐dimensional hydrogen‐bonded structures.     

Symmetric Periodic Orbits and Uniruled Real Liouville Domains

A real Liouville domain is a Liouville domain with an exact anti-symplectic involution. The authors call a real Liouville domain uniruled if there exists an invariant finite energy plane through every real point. Asymptotically, an invariant finite energy plane converges to a symmetric periodic orbit. In this note, they work out a criterion which guarantees uniruledness for real Liouville domains.