On the local linear independence of translates of a box spline
The maximum M of a critical Bienaymé-Galton-Watson process conditioned on the total progeny N is studied. Imbedding of the process in a random walk is used. A limit theorem for the distribution of M as N → ∞ is proved. The result is trasferred to the non-critical processes. A corollary for the maximal strata of a random rooted labeled tree is obtained.
A nearly sharp lower bound on the length of the longest trail in a graph on n vertices and average degree k is given provided the graph is dense enough (k ≥ 12.5).