Re: Example...
> From: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
> To: "Dan Zuras Intervals" <intervals08@xxxxxxxxxxxxxx>
> Subject: Example...
> Date: Wed, 11 May 2011 14:40:27 -0500
>
> > I'm not sure if this suggests that the decoration should
> > be the best (safest) of the input decorations as Nate &
> > John have discussed. But it is a reasonable suggestion.
> >
> > I would certainly agree in the case of union (something
> > that seeks the truth about EITHER of its operands) but
> > I'm a bit more iffy about intersection.
> >
> > Let me ask of the assembled lurkers if there is some
> > optimization problem that seeks to find a point (or
> > interval) where two distinct things are true. Something
> > for which the natural interval extension might be
> > thing1 \intersect thing2. Something more revealing
> > than the Newton's example.
> >
> > Can anyone present us with such a problem to serve as
> > a guide in answering the decoration question?
> >
>
> Hi Dan,
>
> FWIW, here is an example about the intersection operation, posted originally
> to P1788 on 1/7/2011.
>
> Union may need a little more thought, but it seems clear to me that for min,
> max and intersection the worst decoration is always required.
>
> Nate
>
An excellent example Nate.
And your figure 4 suggests we need to be conservative
on our assignment of decorations for intersection to
prevent false positives.
I'm not sure what needs to be done to prevent false
negatives.
BTW, since both f() & g() are defined for expressions
which are positive one could make this example less
contentious by replacing them with some f'() & g'()
which are defined everywhere but are only positive
somewhere.
Then the problem becomes: find subsets such that f'()
& g'() are both positive.
For that matter, one can generalize your example to
test matters of continuity as well by replacing the
operands with f''() = signum(f'()) & g''() =
signum(g'()) & seeking subsets where both of these
are positive (the boundaries of which are
discontinuous).
Does any of this illuminate the correct rule for
decorating intersections?
Dan
P.S. - A moment's reflection shows that your example
could be used to test union by asking where EITHER
function is defined. Only a small sliver of the
circle in figure 3 would remain undefined for both.
How must the decoration rule work in this case?