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On-line arbitrarily vertex decomposable trees

A tree T is arbitrarily vertex decomposable if for any sequence τ of positive integers adding up to the order of T there is a sequence of vertex-disjoint subtrees of T whose orders are given by τ. An on-line version of the problem of characterizing arbitrarily vertex decomposable trees is completely solved here.     

Orthonormal polynomial bases in function spaces

We construct polynomial orthonormal bases in various function spaces. Our bases have linear order of growth of degrees of polynomials. We show that this order is optimal.     

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Ueberblick über das Gebiet der Gallerten und Gele

Ohne ZusammenfassungStark gekürzt vorgetragen.     

On the Neighbour-Distinguishing Index of a Graph

A proper edge colouring of a graph G is neighbour-distinguishing provided that it distinguishes adjacent vertices by sets of colours of their incident edges. It is proved that for any planar bipartite graph G with Δ(G)≥12 there is a neighbour-distinguishing edge colouring of G using at most Δ(G)+1 colours. Colourings distinguishing pairs of vertices that satisfy other requirements are also considered.     

Luminescence Properties of Tb-Doped Yttrium Disilicate Prepared by the Sol–Gel Method

A sol–gel method for the synthesis of nanosized powders of yttrium disilicate doped with Tb^{3 +} is presented. The influence of grain size and annealing temperature on the structure and luminescence properties of the materials obtained has been investigated. The preparation of glass-ceramic composite has been also described. The effect of incorporating the nanocrystals into sol–gel silica glasses on their physicochemical properties has been shown and discussed. A lack of influence of the annealing above 1300°C on the lifetimes measured for the nanocrystals embedded in the glass, as compared to the Y₂Si₂O₇:Tb^{3 +} powders, has been observed.