m at h . R A ] 2 5 Fe b 20 00 – 1 – BIRKHOFF ’ S THEOREM FOR PANSTOCHASTIC MATRICES

ar X iv : m at h / 99 11 10 9 v 1 [ m at h . R A ] 1 5 N ov 1 99 9 – 1 – BIRKHOFF ’ S THEOREM FOR PANSTOCHASTIC MATRICES

The panstochastic analogue of Birkhoff's Theorem on doubly-stochastic matrices is proved in the case n = 5. It is shown that this analogue fails when n > 1, n = 5.

Birkhoff's Theorem for Panstochastic Matrices

ar X iv : m at h / 99 07 08 5 v 3 [ m at h . G R ] 2 5 Fe b 20 00 LOOPS AND SEMIDIRECT PRODUCTS

. at om - p h ] 1 5 Fe b 20 00 Precision Measurement of the Lifetime of the 3 d 2 D 5 / 2 state in 40 Ca +

Operators Associated with a Pair of Nonnegative Matrices

Let Amx„, Bmxn, Xnxl, and Ymxl be matrices whose entries are nonnegative real numbers and suppose that no row of A and no column of B consists entirely of zeroes. Define the operators U, T and T by (UX)t-X? [or (UY),= Y;1], T=UB'UA and T' = UAUB'. Tis called irreducible if for no nonempty proper subset S of (1, • ■ ■ , n} it is true that X,=0, ieS; X,^0, i$ S, implies (TX),=0, ieS; (TX)i^O, i $...