dq.(m) 1 d[(-) dOo()
dq~i) OC2 I () (a) d(( (c) () Eq. 7.3
with nonzero denominator polynomial 6(a)2. The following proposition is a conse-
quence.
Proposition 7.3: For all a > 0, the components of the first derivative of q, with respect to
a are continuous rational functions of a.
7.3 The Absorbing State Rows of IP-Il and I P,-II
The rows corresponding to one-operator absorbing states in the determinant
SP- II have a particularly simple form. The nondiagonal elements of row nA e S,', which
represents a uniform population of solutions iA e S, are given by Eq. 4.19 and 4.29
respectively for the two and three-operator cases. The principal diagonal element is
obtained by evaluating Eq. 4.19 or 4.29 at m = nA and subtracting 1. Thus,
(M MHU(i ,i )
(nA (I+ O)M
S-1
(1 + a)ML
1 - MLao- ML(ML l)a2/2 -. O-a"M
(1 + a)"L
-MLa + O(ao2)
(1 + )ML
and if the general element in | P- I is denoted by T(m I n), then the elements of row nA
can be written as
-MLa+ O(a2)
S--- +--[ n = n,
(1 + a) Eq. 7.4
Vn E S,': T(n I|n)= o )(),,),^i Eq. 7.4
ne S'- {nA}
n(1 + a)'-