On Lipschitz and D.C. surfaces of finite codimension in a Banach space

Properties of Lipschitz and d.c. surfaces of finite codimension in a Banach space and properties of generated σ-ideals are studied. These σ-ideals naturally appear in the differentiation theory and in the abstract approximation theory. Using these properties, we improve an unpublished result of M. Heisler which gives an alternative proof of a result of D. Preiss on singular points of convex functions.     

Zur Methodik der Chlorveraschung

Analytical and Bioanalytical Chemistry -     

Crown reductions for the Minimum Weighted Vertex Cover problem

The paper studies crown reductions for the Minimum Weighted Vertex Cover problem introduced recently in the unweighted case by Fellows et al. [Blow-Ups, Win/Win's and crown rules: some new directions in FPT, in: Proceedings of the 29th International Workshop on Graph Theoretic Concepts in Computer Science (WG’03), Lecture notes in computer science, vol. 2880, 2003, pp. 1-12, Kernelization algorithms for the vertex cover problem: theory and experiments, in: Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX), New Orleans, Louisiana, January 2004, pp. 62-69]. We describe in detail a close relation of crown reductions to Nemhauser and Trotter reductions that are based on the linear programming relaxation of the problem. We introduce and study the so-called strong crown reductions, suitable for finding (or counting) all minimum vertex covers, or finding a minimum vertex cover under some additional constraints. It is described how crown decompositions and strong crown decompositions suitable for such problems can be computed in polynomial time. For weighted König-Egerváry graphs (G,w) we observe that the set of vertices belonging to all minimum vertex covers, and the set of vertices belonging to no minimum vertex covers, can be efficiently computed.Further, for some specific classes of graphs, simple algorithms for the MIN-VC problem with a constant approximation factor r<2 are provided. On the other hand, we conclude that for the regular graphs, or for the Hamiltonian connected graphs, the problem is as hard to approximate as for general graphs.It is demonstrated how the results about strong crown reductions can be used to achieve a linear size problem kernel for some related vertex cover problems.     

Erratum

Ohne Zusammenfassung     

Enantioseparations of non-benzenoid and oligo-Tröger’s bases by HPLC on Whelk O1 column

The separation of enantiomers of several ‘bis- and tris-Tröger’s bases’ by HPLC on commercially available chiral stationary phase Whelk O1 is described for the first time. The observed structure–enantioselectivity relationships are in agreement with the previously established molecular recognition model. For all ‘bis- and tris-Tröger’s bases’ studied, satisfactory to excellent enantioselectivities were observed.     

Simple computer solution of coupled power equations

A numerical method for solution of the coupled power equations in multimode optical fibres is presented. The method is based on the assumption that the contribution of weak mode coupling can be concentrated into discrete points along the fibre. The matrix form of the solution is derived, the accuracy of the method is briefly discussed and an example of practical results is also shown.