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N}, 0 < #5 < #R, such that H -> Ae -» S. Then for all e""1 e £ n _i, A en -i ftos a positive column. Proof. ,n} and choose any e™"1 e -En-i. Apply the hypothesis to the set HI and the matrix Afl. Thus there is a set 7^2 such that KI —> Aei —> 7^2 and #7^2 < n — 1. Successively applying our hypothesis produces sets Kj -> A £j -* TZj+i, j = 1,... ,n - 1 with 0 < #7t,-+i < #7lj - 1. n = {i}. Therefore, we conclude that the ith column of Atl • • • Atn_1 = Aen-i is positive. 2. Let Af,c e {0,1}, be n x n nonnegative matrices such that either or then AQTI-I has a positive column.

C) The last row of A0 is the same as the first row of AI. 81) and forces the repetition of the nth control point of the new control polygon D. (d) AQ and A\ are nonsingular. Let us prove this fact. Suppose A0x = 0 for x e R n . Then the first n components of the vector Ax are zero and, likewise, the first n components of £2n-l,n-l . . fln+l,lx are zeroThematrb,£2n-l,n-l leaveg the first n _ 2n 2 n 2 n+l l 1 components of y := B " ' " • • •B ' x unchanged and so they must be zero. Since the nth component of B2n~1'n~ly is zero and is also a strict convex combination of the n — 1th and nth component of y we see that actually the first n components of y are zero.

Tk)T does not have a positive column. To make use of this information we define for every matrix C with nonnegative elements its signum, viz. rri There are m distinct signum matrices of order n. , m, cannot exhaust all possible signum matrices since they all do not have a positive column. 74), it follows inductively that for £ = 1,2,... 74), applied to the matrices A and A^B. \ • • • fr e r +i • • • es fr+i • • • es • • •. That is. beyond the first r binary digits of to, the remaining digits repeat with period s — r.