Self-intersections in combinatorial topology: statistical structure

Oriented closed curves on an orientable surface with boundary are described up to continuous deformation by reduced cyclic words in the generators of the fundamental group and their inverses. By self-intersection number one means the minimum number of transversal self-intersection points of representatives of the class. We prove that if a class is chosen at random from among all classes of m letters, then for large m the distribution of the self-intersection number approaches the Gaussian distribution. The theorem was strongly suggested by a computer experiment with four million curves producing a very nearly Gaussian distribution.     

Zum Nachweis des Titans und Cyans

Ohne Zusammenfassung     

The Dynamical Problem for a Rigid Body in Special Relativity

The dynamical problem for a continuous body constrained to undergo only plane BORN motions is solved in special relativity.     

New hybrid [2]catenanes based on a 4,4'-bipyridinium ligand

The self-assembly-mediated synthesis of metallomacrocycles 4a and 4b from (en)M(NO3)2 (M = Pd, Pt) and bipyridinium ligand 3 is described. The reaction is templated by disodium p-phenyldiacetic dicarboxylate, which is inserted into the macrocyclic cavity. Similarly, the self-assembly process between ligand 3, (en)M(NO3)2 (M = Pd, Pt), and the macrocyclic polyether 6 resulted in the formation of hybrid catenanes 7a and 7b. In the [2]catenanes, the circumrotation of the macrocyclic polyether through the cavity of the metallocycle is slow on the 1H NMR time scale.