Let a = ([a.sub.1], ..., [a.sub.n]), r = ([r.sub.1], ..., [r.sub.m]) and s = ([s.sub.1], ..., [s.sub.m]) be sequences of

nonnegative integers with [a.sub.1] + ...

Corollary 2 Let l and n be

nonnegative integers, [2.sup.l] < n, and let 0 [less than or equal to] r < [2.sup.l+1] be such that n [equivalent to] r (mod [2.sup.l+1]).

The proof is divided into the following cases by considering r and l such that m = 8r + l where r is a

nonnegative integer and l [member of] {0,1, ..., 7}.

such that [f.sup.{k)](y(t),t) are bounded in [y([t.sub.0]) - [theta], y([t.sub.0]) + [theta]] x [[t.sub.0], [[bar.t].sub.J]] for k = 0,1, ..., l + 1 for some

nonnegative integer l [less than or equal to] n and some [theta] [member of] [bar.B]([[theta].sub.0], R), where [bar.B]([theta],R) = {z [member of] [R.sup.n]: [parallel]z - [[theta].sub.0][parallel] [less than or equal to] R] for some positive real R with [x.sup.(j)]([t.sub.0]) = [y.sup.(j)]([t.sub.0]) for j = 0,1, ..., l + 1.

This new basis has nice properties, as the fact that its multiplication table contains only

nonnegative integers.

where k is a

nonnegative integer, n and m are natural numbers.

If dim(X) is finite, then there exists a

nonnegative integer n such that dim(X) = n.

for all a, b [member of] A and each

nonnegative integer n; it is siad to be strongly if d0 = I.

We need to show that there exist

nonnegative integers [n.sub.i] for i = 0, ..., l and integers [mathematical expression not reproducible].

Algorithm 1 Let n be a

nonnegative integer and [lambda] [member of] C.

Then h(c) is defined to be the minimal

nonnegative integer i such that [mathematical expression not reproducible] is congruent modulo n to a cycle vertex in G(n, k).

where [l.sub.1] := deg Q' is a

nonnegative integer. Then, combining (35) and (41) yields