Eine ErweiterungT(V′) des Ordinalzahlensystems 58-0158-0158-01(Λ0) von G. Jäger

Summary This paper gives a recursive generalization of a strong notation system of ordinals, which was devellopped by Jäger [3]. The generalized systemT(V) is based on a hierarchy of Veblen-functions for inaccessible ordinals. The definition ofT(V) assumes the existence of a weak Mahlo-ordinal. The wellordering ofT(V) is provable in a formal system of second order arithmetic with the axiom schema of ₂ ¹ -comprehension in a similar way, as it is proved in [6] for the weaker notation systemT(V).     

Stochastic modeling of a billiard in a gravitational field: Power law behavior of Lyapunov exponents

We consider the motion of a point particle (billiard) in a uniform gravitational field constrained to move in a symmetric wedge-shaped region. The billiard is reflected at the wedge boundary. The phase space of the system naturally divides itself into two regions in which the tangent maps are respectively parabolic and hyperbolic. It is known that the system is integrable for two values of the wedge half-angle ₁ and ₂ and chaotic for ₁<< ₂. We study the system at three levels of approximation: first, where the deterministic dynamics is replaced by a random evolution; second, where, in addition, the tangent map in each region is, replaced by its average; and third, where the tangent map is replaced by a single global average. We show that at all three levels the Lyapunov exponent exhibits power law behavior near ₁ and ₂ with exponents 1/2 and 1, respectively. We indicate the origin of the exponent 1, which has not been observed in unaccelerated billiards.     

The synthesis of N-substituted ureas II: Nucleophilic substitution of ureas at the carbonyl group

Summary N-Arylsubstituted ureas undergo exchange of the N-residue upon reaction with amines. Using kinetic measurements, investigation of product distribution, regioselectivity, catalysis, and substrate influences, it was shown that this reaction proceeds via a second order nucleophilic substitution at the urea carbonyl center. By means of semiempirical calculations using the MNDO method the alternative mechanism of fragmentation was investigated.

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Zur Synthese N-substituierter Harnstoffe II: Nucleophile Substitution von Harnstoffen an der Carbonylgruppe

Zusammenfassung N-Arylsubstituierte Harnstoffe erleiden bei der Reaktion mit Aminen einen Austausch des N-Restes. Durch kinetische Messungen, Untersuchung der Produktverteilung, Regioselektivität, Katalyse und des Einflusses des Edukts konnte abgeleitet werden, daß es sich hiebei um eine nucleophile Substitution zweiter Ordnung am Carbonylzentrum des Harnstoffs handelt. Durch semiempirische Rechnungen mit Hilfe der MNDO Methode wurde der alternative Reaktionsmechanismus einer Fragmentierung untersucht.