Policies are used to control the behavior of large-scale systems. After policies are designed and before policies are implemented, it should be possible to analyze the policies and verify that they behave as expected. Graph transformations provide a good method to carry out the analysis and verification because of the intuitive representation of policies and the mature theory. Here, we show how...

A phase-field approach for quantitative simulations of grain growth in anisotropic systems is introduced, together with a new methodology to derive appropriate model parameters that reproduce given misorientation and inclination dependent grain boundary energy and mobility in the simulations. The proposed model formulation and parameter choice guarantee a constant diffuse interface width and co...

Note that if ∑ agg ∈ I, then ∑ ag = 0 so that ∑ agg = ∑ agg− ∑ ag1 = ∑ ag(g−1), so that every element in I is a Z-linear combination of the elements g− 1 (g ∈ G, g 6= 1). Also, if ∑ ag(g− 1) = 0, it follows that ag = 0 for all g so that the elements g − 1 (g ∈ G, g 6= 1) are linearly independent. (b) Notice that the ideal generated by the elements s− 1 (s ∈ S) is the same as the ideal generated...