guaranteed cost control of discrete linear systems.
Let be a square -matrix. Then is a Hall matrix provided it has a nonzero permanent. The Hall exponent of is the smallest positive integer , if such exists, such that is a Hall matrix. The Hall exponent has received considerable attention, and we both review and expand on some of its properties. Viewing as the adjacency matrix of a digraph, we prove several properties of the Hall exponents of line digraphs with some emphasis on line digraphs of tournament (matrices).
It is an open question whether Nehari's theorem on the circle group has an analogue on the infinite-dimensional torus. In this note it is shown that if the analogue holds, then some interesting inequalities follow for certain trigonometric polynomials on the torus. We think these inequalities are false but are not able to prove that.
A well known theorem of Nehari asserts on the circle group that bilinear forms in H² can be lifted to linear functionals on H¹. We show that this result can be extended to Hankel forms in infinitely many variables of a certain type. As a corollary we find a new proof that all the norms on the class of Steinhaus series are equivalent.