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Re: 11 year olds homework
Posted:
Nov 16, 2004 8:04 PM
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On 14 Nov 2004 06:31:37 GMT, israel@math.ubc.ca
(Robert Israel) wrote:
>Let F be the mapping (a,b,c,d) -> (|a-b|,|b-c|,|c-d|,|d-a|) on R^4.
>Note that F is positively homogeneous (i.e. F(tx) = t F(x) if
>t >= 0), so multiplying a vector by a positive scalar doesn't
>change the number of iterations needed to reach (0,0,0,0).
>[...] this mapping has arbitrarily long orbits (on Z^4 as
>well as R^4).
Even worse:
If k is the unique real solution of k^3 + 2k^2 - 2 = 0,
and u is the vector (1, k + 1, (k + 1)^2, k^2 + 3k + 3),
then F(u) = (k, k(k + 1), k + 2, k^2 + 3k + 2) = ku
(because k(k + 1)^2 = k + 2), therefore F^n(u) = (k^n)u
\neq (0, 0, 0, 0) for all n >= 0.
--
Angus Rodgers
(angus_prune@ eats spam; reply to angusrod@)
Contains mild peril
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