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Re: unwarranted assumption?

• To: mathgroup at yoda.physics.unc.edu
• Subject: Re: unwarranted assumption?
• From: rubin at msu.edu (Paul A. Rubin)
• Date: Tue, 28 Jun 94 12:00:04 EDT

>Dear Mathgroup
>
>The following expression
>
>Simplify[x*D[x^(1/3)/(k+x^(1/3)), x]/((x^(1/3)/(k+x^(1/3)))*
>(1-(x^(1/3)/(k+x^(1/3)))))]
>
>gives the possibly surprising and rather elegant result
>
>Out [] =
>
>   1
>   -
>   3
>
>[Turns out that this expression approximates dependence of cloud
>reflectivity on number concentration of cloud drops]
>
>I was surprised however that mathematica didn't choke on this expression.
>My question is how does mathematica know that k is not zero?

It doesn't.

>I would have
>expected mathematica to consider all possible values of k and at least flag
>the possibility.

Actually, in doing algebraic manipulations, I think it tends to cancel
factors without concern for zero/zero - which isn't all that bad, really (if
you assuming that the final 1/3 is really 1/3 times a bunch of fractions of
the form expr/expr, where numerator and denominator are the same expression,
than if any denominators go to zero the numerators go at the same rate, and
the limit is 1.  (Or did I just commit some freshman error in that logic?)

>Feel free to knock me over the head if this is a trivial question.  I was
>kinda surprised that mma gave me the answer I "expected" instead of a bunch
>of red flags.
>
>Generally my experience with mathematica is the contrary; that I would have
>had to have told it that k was not zero.

Sometimes yes, sometimes no.  For instance, Reduce[] typically gives a
"logic tree" of solutions contingent on assumptions, while Solve[] just
gives a "generic" solution.  Some simplifications are not done unless
manually forced, suggesting that Mma was reluctant to do it because there is
a possible pitfall.  Then again, on sci.math.symbolic there has been some
flaming of Mma for giving Integrate results that didn't consider all
possible parameter values.

>
>mathematically yours  -steve
>
>
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* Paul A. Rubin                                  Phone: (517) 336-3509   *
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Mathematicians are like Frenchmen:  whenever you say something to them,
they translate it into their own language, and at once it is something
entirely different.                                    J. W. v. GOETHE