The Urbanik generalized convolutions in the non-commutative probability and a forgotten method of constructing generalized convolution

The paper deals with the notions of weak stability and weak generalized convolution with respect to a generalized convolution, introduced by Kucharczak and Urbanik. We study properties of such objects and give examples of weakly stable measures with respect to the Kendall convolution. Moreover, we show that in the context of non-commutative probability, two operations: the q-convolution and the (q,1)-convolution satisfy the Urbanik??s conditions for a generalized convolution, interpreted on the set of moment sequences. The weak stability reveals the relation between two operations.     

Data Dependent Triangulations for Piecewise Linear Interpolation

Given a set of data points in R² and corresponding data values,it is clear that the quality of a piecewise linear interpolationover triangles depends on the specific triangulation of thedata points. While conventional triangulation methods dependonly on the distribution of the data points in R² in this paperwe suggest that the triangulation should depend on the datavalues as well. Several data dependent criteria for definingthe triangulation are discussed and efficient algorithms forcomputing these triangulations are presented. It is shown fora variety of test cases that data dependent triangulations canimprove significantly the quality of approximation and thatlong and thin triangles, which are traditionally avoided, aresometimes very suitable.