On qualitatively independent partitions and related problems

The paper is concerned with several related combinatorial problems one of which is that of estimating the numbers of qualitatively independent p-partitions. Besides nonconstructive basic estimates, a constructive procedure yielding not much worse ones is presented. In conclusion, some applications are shown.     

Eine Bemerkung über volle Einbettungen von Kategorien von Algebren

Ohne ZusammenfassungUnterstützt von der Alexander von Humboldt-Stiftung.     

On classes of relations and graphs determined by subobjects and factorobjects

The classes of relations and graphs determined by subobjects and factorobjects are studied. We investigate whether such classes are closed under products, whether they are finitely generated by products and subobjects and whether a class can be described alternatively by subobjects and factorobjects. This is related to good characterizations.     

Resonant Homoclinic Flip Bifurcations

This paper studies three-parameter unfoldings of resonant orbit flip and inclination flip homoclinic orbits. First, all known results on codimension-two unfoldings of homoclinic flip bifurcations are presented. Then we show that the orbit flip and inclination flip both feature the creation and destruction of a cusp horseshoe. Furthermore, we show near which resonant flip bifurcations a homoclinic-doubling cascade occurs. This allows us to glue the respective codimension-two unfoldings of homoclinic flip bifurcations together on a sphere around the central singularity. The so obtained three-parameter unfoldings are still conjectural in part but constitute the simplest, consistent glueings.     

Less Is More: Three‐Coordinate C,N‐Chelated Distannynes and Digermynes

We report here the synthesis of new C,N‐chelated chlorostannylenes and germylenes L³MCl (M=Sn( 1 ), Ge ( 2 )) and L⁴MCl (M=Sn( 3 ), Ge ( 4 )) containing sterically demanding C,N‐chelating ligands L^{3, 4} (L³=[2,4‐di‐tBu‐6‐(Et₂NCH₂)C₆H₂]^?_; L⁴=[2,4‐di‐tBu‐6‐{(C₆H₃‐2′,6′‐iPr₂)N=CH}C₆H₂]^?). Reductions of 1 – 4 yielded three‐coordinate C,N‐chelated distannynes and digermynes [L^{3, 4}M ]₂ for the first time ( 5 : L³, M=Sn, 6 : L³, M=Ge, 7 : L⁴, M=Sn, 8 : L⁴, M=Ge). For comparison, the four‐coordinate distannyne [L⁵Sn]₂ ( 10 ) stabilized by N,C,N‐chelate L⁵ (L⁵=[2,6‐{(C₆H₃‐2′,6′‐Me₂)N?CH}₂C₆H₃]^?) was prepared by the reduction of chlorostannylene L⁵SnCl ( 9 ). Hence, we highlight the role of donor‐driven stabilization of tetrynes. Compounds 1 – 10 were characterized by means of elemental analysis, NMR spectroscopy, and in the case of 1 , 2 , 5 – 7 , and 10 , also by single‐crystal X‐ray diffraction analysis. The bonding situation in either three‐ or four‐coordinate distannynes 5 , 7 , and 10 was evaluated by DFT calculations. DFT calculations were also used to compare the nature of the metal–metal bond in three‐coordinate C,N‐chelating distannyne [L³Sn]₂ ( 5 ) and related digermyme [L³Ge]₂ ( 6 ).