Abstract:
This thesis is devoted to investigation of some properties of the permanent function over the set Omega_n of n-by-n doubly stochastic matrices. It contains some basic properties as well as some partial progress on Foregger's conjecture.
CONJECTURE[Foregger]
For every n\in N, there exists k=k(n)>1 such that, for every matrix A\in Omega_n,
per(A^k)<=per(A).
In this thesis the author proves the following result.
THEOREM
For every c>0, n\in N, for all sufficiently large k=k(n,c), for all A\in\Omega_n which minimum nonzero entry exceeds c,
per(A^k)<=per(A).
This theorem implies that for every A\in\Omega_n, there exists k=k(n,A)>1 such that
per(A^k)<=per(A).
