setting row mA to T'. Further, by virtue of their construction (Eq. 7.7 and 7.10), the single
row dependence of the matrix elements on the conditional probability array [P(i I n)]
employed in developing the results in Section 9.3 for P and Pi applies to P(mA)' and Pi'
as well. Thus, Eq. 9.26-27 should extend with very little modification to the determinants
- Pi^'-1' = IP(mA)'--I' P' -I' whose zero mutation probability limits are
required by Propositions 7.5-6. The following paragraphs highlight the required modifi-
cations and employ the result to examine two simple examples.
In the ac --- 0 counterpart of Eq. 9.26-27, m is limited to membership in the set of
one-operator absorbing states (i.e. m = mA e SA'), a consequence of which is that
f r 1.
mA)= .
Also, all rows of the determinant other than mA correspond to nonabsorbing states (i.e.
n e K c S' SA'). Thus, the determinant order is N' N + 1 and the differentiation index
array is order (N' -N + 1) x N with rows corresponding to row indices
n e S' SA' + {mA. The rows ofr are limited to rj = 0 and rE, S" for n e S' SA. Fur-
ther, if r indicates nonzero order differentiation of any rows which are adjacent to one-
operator absorbing states, then the associated columns of the combinatorial determinant
must reflect the coefficient contribution from the adjacent absorbing state. Thus, if
Co'(mA,r) denotes the limiting counterpart of C'(m,r) in Eq. 9.27 and if
K = {n, n2, -,nk} c S' SA' where (in the state adjacency notation introduced in Section
7.3) n, e S(nA)' # S(mA)' and where n2, -,nk all satisfy nj e SA", then the coefficient of
the order k monomial term uniquely identified by r is given by