ABSTRACT The authors continue their study of extremal problems of Turan type for directed graphs with multiple edges, now permitting any finite non-negative integer multiplicity. Having proved earlier (for the case of multiplicity at most 1) that there exists , for any family of “sample” digraphs, a matrix which represents the structure of an “asymptotically extremal sequence” of digraphs (containing none of the sample digraphs, and having a total number of arcs asymptotic to the maximum), they address themselves to the inverse problem: is every matrix so realized for some finite family of sample digraphs? They prove that this is indeed true for “dense” matrices - having certain integer entries, and such that an associated quadratic form attains its maximum for the standard simplex uniquely at an interior point.
The Tur\'an density \pi(H) of a family H of k-graphs is the limit as n tends to infinity of the maximum edge density of an H-free k-graph on n vertices. Let I^k consist of all possible Tur\'an densities and let F^k be the set of Tur\'an densities of finite k-graph families. ::: Here we prove that F^k contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that F^k contains an irrational number for each k\ge 3. ::: Also, we show that I^k has cardinality of the continuum. In particular, I^k is not equal to F^k.
The oxidative polymorphism of debrisoquine (DBQ) has been determined in 89 patients with colo-rectal cancer and in 556 normal control subjects. Four patients and 34 controls, with a metabolic ratio >12.6, were classified as poor metabolisers of DBQ (n.s.).