Digraph extremal problems, hypergraph extremal problems, and the densities of graph structures

We consider extremal problems 'of Turan type' for r-uniform ordered hypergraphs, where multiple oriented edges are permitted up to multiplicity q. With any such '(r, q)-graph' G^n we associate an r-linear form whose maximum over the standard (n - 1)-simplex in R^n is called the (graph-) density g(G^n) of G^n. If ex(n, L) is the maximum number of oriented hyperedges in an n-vertex (r, q)-graph not containing a member of L, lim"n"->"~ ex(n, L)/n^r is called the extremal density of L. Motivated, in part, from results for ordinary graphs, digraphs, and multigraphs, we establish relations between these two notions.