where (n - 1)' = [[summation].sup.k.sub.i-1] in [.sub.i][2.sup.i-1] if n - 1 has the dyadic expansion [[summation].sup.k.sub.i-0] in [.sub.i][2.sup.i].
Write [mathematical expression not reproducible] the dyadic expansion of n.
If n = 1, it is clear by Fact 2.1 (i), and so we assume n [greater than or equal to] 2 and n - 1 [greater than or equal to] 1 has the dyadic expansion [[summation].sup.k.sub.i-0] in [.sub.i][2.sup.i], with [n.sub.k] = 1.
Let [alpha] = (0.[[alpha].sub.1][[alpha].sub.2] ...) 2 be the unique dyadic expansion of a with infinitely many 0s.
From now, we rephrase the statement of Lemma 8 in terms of the dyadic expansion of [square root of 2].
Regarding the quantities [r.sub.inf] (b) and [r.sub.sup] (b), it has been (even implicitly) conjectured that [r.sub.inf] (b) = [r.sub.sup](b) = 1/2, that is, the asymptotic occurrence rates of 0s and 1s in the dyadic expansion of [square root of 2] coincide with each other.
Long regarded as the master in infant psychology, Tronick (pediatrics, Harvard Medical School) has compiled 36 of his articles on neurobehavior, culture, infant social and emotional interaction, natural and experimental perturbations, and the dyadic expansion
of consciousness and making of meaning.
We also use the frequency derivatives of the Green's function and the Green's dyadic expansions