Editorial

Mathematische Semesterberichte -     

Thomas de Padova: Alles wird Zahl: Wie sich die Mathematik in der Renaissance neu erfand

Mathematische Semesterberichte -     

Stereoselective Ring Opening of Electronically Excited Cyclohexa-2, 4-dienones: Cause and effect

The two conformers of a cyclohexa-2, 4-dienone with different substituents at C(6) on irradiation are believed to undergo ring opening stereospecifically affording a mixture of two configurationally isomeric diene-ketenes (and descendents thereof)- Exceptions are generally found for those dienones with one C and one O substituent or even with two C substituents, if one of them carries a polar group at a site able to interact through space with the ring C?O group. In these cases, only one of the two anticipated diene-ketenes (and descendents thereof) is produced. A thorough investigation of the photochemistry of a series of structurally different cyclohexa-2, 4-dienones on analytical as well as on preparative scale extends our mechanistic knowledge of the various routes from diene-ketenes into a variety of compound classes. Novel compound classes accessible to diene-ketenes are seven-membered carbocycles (by intramolecular aldolization of the zwitterion of appropriately substituted, transiently formed diene-(N, O)-ketene acetals) and β-lactams (by Staudinger reaction).     

The Geodesics of Metric Connections with Vectorial Torsion

The present note deals with the dynamics of metric connections with vectorial torsion, as already described by E. Cartan in 1925. We show that the geodesics of metric connections with vectorial torsion defined by gradient vector fields coincide with the Levi-Civita geodesics of a conformally equivalent metric. By pullback, this yields a systematic way of constructing invariants of motion for such connections from isometries of the conformally equivalent metric, and we explain in as much this result generalizes the Mercator projection which maps sphere loxodromes to straight lines in the plane. An example shows that Beltrami's theorem fails for this class of connections. We then study the system of differential equations describing geodesics in the plane for vector fields which are not gradients, and show among others that the Hopf–Rinow theorem does also not hold in general.     

Coupling artificial actin cortices to biofunctionalized lipid monolayers

We report the assembly of protein supramolecular structures at an air-water interface and coupling of artificial actin cortices to such structures. The coupling strategies adopted include electrostatic binding of actin to monolayers doped with lipids, exposing positively charged poly(ethylene glycol) headgroups; binding of biotinylated actin to lipids carrying biotin headgroups through avidin; binding of actin to membranes through biotinylated hisactophilin (a cellular actin-membrane coupler) using an avidin-biotin linkage; and coupling of actin to membranes carrying chelating lipids through a 15-nm-diameter protein capsid (bacterial lumazine synthase or LuSy) exhibiting histidine tags (which bind both to actin and to the chelating lipid). The distribution of the proteins in a direction normal to the interface was measured by neutron reflectivity under different conditions of pH and ionic strength. In the case of the first three binding methods, the thickness of the actin film was found to correspond to a single actin filament. Multilayers of actin could be formed only by using the multifunctional LuSy couplers that exhibit 60 hexahistidine tags and can thus act as actin cross-linkers. The LuSy-mediated binding can be reversibly switched by pH variations.