(18)
where
Q ≡ [(z/my)2-1/y[KC/(KC - K)]1/y].
The transcendental function LambertW(x) is defined by the relation
LambertW (x) exp(LambertW (x)) = x
and is found in the computer algebra program Mathematica, where it is
called the ProductLog. It arises in the theory of random networks and in
renormalization strategies for quantum field theories.
Fixing KC , m and z, and tuning y again gives behavior recognizably simi-
lar to the simple development of surrounding equatuin (7) above, an exercise
likewise best carried out through a symbolic mathematics program.
An asymptotic relation for f (R), rising toward a finite limit with increase
in R, would be of particular biological interest, implying that ‘language rich-
ness’ increases to a limiting value with population growth, in a loose sense.
Such a pattern is broadly consistent with calculations of the degree of allelic
heterozygosity as a function of population size in the context of a balance
between genetic drift and neutral mutation [27. 41]. Taking
f(R) = exp[m(R - 1)/R]
(19)
gives a system which begins at 1 when R = 1, and approaches the asymp-
totic limit exp(m) as R → ∞. Mathematica 4.2 finds
R = my/z
C LambertW [L],
28
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