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Re: "Assuming"
*To*: mathgroup at smc.vnet.net
*Subject*: [mg85858] Re: "Assuming"
*From*: "Mariano Suárez-Alvarez" <mariano.suarezalvarez at gmail.com>
*Date*: Mon, 25 Feb 2008 07:37:45 -0500 (EST)
*References*: <20080221171506.200$2n_-_@newsreader.com> <200802221221.HAA08545@smc.vnet.net>
On Feb 23, 7:34 am, Daniel Lichtblau <d... at wolfram.com> wrote:
> Andrzej Kozlowski wrote:
> > On 21 Feb 2008, at 23:15, David W. Cantrell wrote:
>
> >>[Message also posted to: comp.soft-sys.math.mathematica]
>
> >>Andrzej Kozlowski <a... at mimuw.edu.pl> wrote:
> >>[...]
>
> I intersperse two comments, based on two posts to this thread.
>
> [From Andrzej Kozlowski:]> Note also one more thing. Suppose we consider a general rational
> > function p[x]/q[x], (or even a rational function in several
> > variables). Should Mathematica be then required always to try to find
> > if p[x] and q[x] do not have common roots in some algebraic (or even
> > transcendental inthe case of several variables) extension of the
> > rationals? This is far from a trivial problem? [...]
>
> > Andrzej Kozlowski
>
> The only removable singularities, in this context, are in fact from
> exact divisors. So polynomial gcd extraction (used in, say, Together[])
> suffices to remove them.
>
> Other cases where numerator and denominator simultaneously vanish will
> give points of indeterminacy. These are not removable.
>
> I'm not sure in what way transcendentals snuck in here. If the
> polynomials have common roots, they are describable as an algebraic set.
> In particular, isolated common roots will be algebraic numbers.
>
> [From David Cantrell:]
>
>
>
> >>[...] But I do know of a case where
> >>Mathematica goes even further, removing a singularity at which the
> >>function
> >>is defined as a number:
>
> >>In[17]:= FullSimplify[UnitStep[-x^2]]
> >>Out[17]= 0
>
> >>despite the fact that correctly
>
> >>In[18]:= UnitStep[-x^2] /. x -> 0
> >>Out[18]= 1
>
> >>Perhaps the simplification above is considered a bug, perhaps not.
>
> >>David
>
> A feature, really. That is, it's wrong, but FullSimplify can make
> mistakes on measure zero sets. We do not generally regard this
> phenomenon as a bug, though we reconsider on case by case basis.
How does that `measure zero' allowance work in a context
of something like
Assuming[Element[x, Integers], FullSimplify[something]]
?
-- m
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