Re: Revised version of Level 1 text (draft)

[Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>]

Arnold Neumaier wrote:
> John Pryce wrote:
> > Arnold, P1788
> >
> > Here are two examples for Arnold's specification of intersection of 
> > decorated intervals, using the (ill<emp<con<def<saf) scheme.
> >
> > Example 1
> > ---------
> > Suppose f(x) = |x| is represented in two (admittedly silly) ways as
> >   g(x) = |2x-x|
> >   h(x) = x sign(x)
> > (where sign(x) is -1 if x<0, 0 if x=0, 1 if x>0).
> > Let xx = [-1,1]. Then I get
> >   gg = g(xx) = abs(2*xx-xx) = abs([-2,2]-[-1,1]) = abs([-3,3]) = [0,3] 
> > decorated dg = saf,
> >   hh = h(xx) = [-1,1]*sign([-1,1] = [-1,1]*[-1,1] =             [-1,1] 
> > decorated dh = def.
> > Each is a valid enclosure of range(f,xx) so let's improve it by taking 
> > the intersection. Arnold's recipe uses the dg>dh case and returns
> >   (ff,df) = (hh,dh) = ([-1,1],def),
> > not very good. Whereas my analysis suggests we should return
> >   (ff,df) = (gg intersection hh, max(dg,dh)) = ([0,1], saf)
> > which happens to be sharp in this case.
>
> Your recipe
>     (ff,df) = (gg intersection hh, max(dg,dh)) = ([0,1], saf)
> is ok here but gives the wrong result if h(x) happens not to be defined 
> everywhere, such as when using
>      g(x)=|x|, h(x)=(sqrt(x))^2
> My formula (corrected formulation below) was constructed in such a way 
> that this case is still correct, which is important for my original 
> example.

As Ned Nedialkov pointed out to me, the example
   g(x)=|x|, h(x)=(sqrt(x))^2
is not appropriate. It doesn't satisfy my stricter criterion below
but only the old version that was too permissive.

Sorry for having created confusion.



> Indeed, here is no theoretical justification for an intersection. So my 
> criterion for applicability should be formulated as follows:
> g and h must be valid expressions for the function of interest, except 
> at isolated values of some argument.