Matrix Theory.

Two areas of matrix theory are discussed the theory of permanents, and the theory of nonnegative matrices. Paper 1 deals with permanental compounds and their use in recurrence formulas for permanents of 0,1-circulants and in related asymptotic formulas. Paper 2 is a extensive survey of the progress in the theory of permanents achieved during the quadrennium 1982-1985. Paper 3 deals with the problem of determining the minimum permanent in the set of n x n doubly stochastic matrices whose first main diagonal entries are equal to zero. The case k o is the famed van der Waerden conjecture. The case k 1 can be easily solved by a method similar to that used by Egorycev in proving the van der Waerden conjecture. For k 2 Egorycevs techniques are of limited use. The case was solved by me in 1984. For 3 or - n the problem is still unsolved.