has a period 4 point, but Theorem C implies that, if / can be extended to an
^-nonexpansive from
with /(0) = 0, / can have at most a period 3
point. A contradiction. Thus the map / defined above, cannot be extended to a
map F :

with F(0) = 0. In general not much seems to
be known about the extension properties of
maps (see [21]).
We conclude this introduction with a short outline of the paper. In Chapter
2 we recall from [13] sufficient conditions that sets S be array-admissible for n,
and in Chapter 3 we further improve these conditions. In Chapter 4 we establish
some basic properties of P{ri) and indicate how these properties allow us explicitly
to compute the sets P(n). In Chapter 5 we study necessary conditions for sets S
to be array-admissible for n, and we prove Theorems A and B. In Chapter 6 we
prove Theorem C. In Chapter 7 we give more precise properties of Q(n), and as a
consequence we obtain Theorem D. In Chapter 8 we give a proof of Theorem 1.5 and
in Chapter 9 we include a proof of our claim for nonnegative matrices mentioned
in the first part of this chapter.
A computer program plays a role in proving Theorem C for 25 n 50. A
description and listing of the program is given in Appendix A and in Appendix B
we include numerical data.
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