Marco, Jürgen, P1788
On 28 Jun 2011, at 21:13, Jürgen Wolff von Gudenberg wrote:
we have prepared a third motion on decorations that clearly follows
the KISS principle. Perhaps it is too simple, but we are convinced
that it works better than the two current approaches. We only have 4
different decorations and one linear order.
We are eager to read your comments.
Perhaps we can prolong the discussion time again? but there is not
much discussion(!)
Thank you. I like the push back towards simplicity. Arnold has
more-or-less completed his new FTDIA proof and has sent me a part for
checking. I think we both want to revert from the present
7-decoration idea, and get rid of boundedness. Yes, please chairman,
can we extend the discussion period?
That said, I have some criticisms.
1. Theorem 1 on p2 is surely well known to be false.
Let f(x)=x for real x. Define F: IR -> IR by
F(xx) = Entire if 0 not in xx, and xx otherwise.
Then for any xx, and x in xx, we have f(x) in F(xx), so F
is an interval extension of f.
But xx=[1,2], yy=[0,2] give F(xx)=Entire, F(yy)=[0,2], so
isotonicity fails.
2. Theorem 2 (1st part) is carelessly stated, in fact meaningless as
written IMO.
It's not "For every interval extension F ...", but "For the
particular interval *function* F that results from applying
straight-
forward interval computation ..., F(xx) encloses the range of f
over xx" -- or better, "... encloses the range over xx of the point
function defined by the expression f."
3. And surely Theorem 2 (2nd part) needs some continuity assumption,
because of the effect of taking the hull. I think it's false for
f(x)=1/x with xx=[-1,1]. And for sign(x) with xx=[0,0.5].
Further, and I trust more constructive, remarks to follow.
John