Recently Howe [4] has considered the oscillation theory for the two-parameter eigenvalue problem

1a

1b

subjected to the boundary conditions

2a

2b

where for i = 1, 2, — ∞<ai<bi <∞, and qi are real-valued, continuous functions in [ai, bi ], pi is positive in [ai, bi z], and pi (ai )=pi (bi ). Furthermore, it is also assumed that (A1B2—A2B1 )≠0 for all values of x 1 and x 2 in their respective intervals.

It is well-known that Wilks’ Λ criterion is distributed as the product of p independent beta variables in the p-variable null-case [3]. In the collinear case, Λ is still distributed as the product of p independent beta variables, one of them following a non-central beta density. Thus when p=2, the exact non-null distribution of Λ in the collinear case is given by the product of two independent beta variables, one central and the other having non-centrality parameter λ.

The problem of finding bounds for the number h(n) of Hamiltonian circuits on the n-cube has been studied by several authors, (1), (2), (3). The best upper bound known is due to Larman (5) who proved that .

In this paper we use a result of Nijenhuis and Wilf (4) on permanents of (0, 1)- matrices to show that for n≥5

where τ, a and c are constants.

Let V(G) and E(G) denote the vertex set and the edge set of a graph G; let Kn denote the complete graph with n vertices and let Kn, m denote the complete bipartite graph on n and m vertices. A Hamiltonian cycle (Hamiltonian path, respectively) in a graph G is a cycle (path, respectively) in G that contains all the vertices of G.