Free web hosting music - 4.6.3 EVALUATION OF POWERS 455 From this definition
4.6.3 EVALUATION OF POWERS 455 From this definition it follows that ej+di–d 2 x > ej+di-d-mmj, for all X E Mij. (40) As an example of this detailed construction, let us consider the star chain 1, 2, 3, 5, 10, 20, 23, for which t = 3, r = 6, d = 3, f = 3. We obtain the following array of multisets: (do,dl,…,dG): 0 1 1 1 1 1 j 2 1 3 3 (ao,al,…,a6): 1 2 20 23 (MO3, Ml3,. . . , M63): M3 e3 = 0, m3 = 1 (Mo2, Ml2,. . . , Ms2): 111: Ad2 e2 = 1, m2 = 1 (MOl,Mll,…,MSl): ., / 0 2 2 MI el = 2, ml = 1 (Moo, MIO,. . .,MGo): 3 3 MO eo = 4, mo = 2 3 3 I SO sl 82 s3 s4 SE, 496 Thus M40 = {2,2}, etc. From the construction we can see that di is the largest element of St; hence 4 E Mio. (41) The most important part of this structure comes from Eq. (40); one of its immediate consequences is Lemma K. If Mz3 and Mu, both contain a common integer x, then -mu < (e3 -e,)-(d,-dd,) < mj. 1 (42) Although Lemma K may not look extremely powerful, it says (when Mij contains an element in common with Mu, and when mj, m, are reasonably small) that the number of doublings between steps u and i is approximately equal to the difference between the exponents k, and ej. This imposes a certain amount of regularity on the addition chain; and it suggests that we might be able to prove a result analogous to Theorem B above, that I*(n) = eo + t, provided that the ej are far enough apart. The next theorem shows how this can be done. Theorem H (W. Hansen, J. fiir die reine und angew. Math. 202 (1959), 129-136). Let n = 2e0 + 2e + ... + 2et, eo > el > .+. > et 2 0. If eo > 2el + 2.271(t -1) and ei-1 2 ei + 2m for 1 5 i 2 t, (43) where m = 2L3.271(t-1)l -t, then l*(n) = eo + t. Proof. We may assume that t > 2, since the result of the theorem is true without restriction on the e s when t 5 2. Suppose that we have a star chain 1 = a0 < a1 < … < a, = n for n with r < eo + t -1. Let the integers d, f, do, . . . , d,,