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Topic: Delta functions.
Replies: 14   Last Post: Apr 22, 2013 8:58 AM

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Waldek Hebisch

Posts: 222
Registered: 12/8/04
Re: Delta functions.
Posted: Apr 15, 2013 5:38 PM
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clicliclic@freenet.de wrote:
>
> clicliclic@freenet.de schrieb:

> >
> > Waldek Hebisch schrieb:

> > >
> > > Using delta functions can lead to problems with zero divisors.
> > > For example in Maple I get:
> > >

> > > > (x - 4)*Dirac(x - 4);
> > > (x - 4) Dirac(x - 4)
> > >

> > > > simplify((x - 4)*Dirac(x - 4));
> > > 0
> > >

> > > > (x - 4)*Dirac(x - 4)/(x-4);
> > > Dirac(x - 4)
> > >
> > > which is clearly inconsistent. I wonder I there is any theory
> > > how to avoid such problems? I mean, what CAS can do to
> > > protect users from wrong results?
> > >

> >
> > Maple's evaluations are consistent applications of the rule
> >
> > f(x)*delta(x-a) -> f(a)*delta(x)

>
> arrrgh: f(x)*delta(x-a) -> f(a)*delta(x-a)
>

> >
> > which defines the meaning of a Dirac delta times a function that is
> > differentiable infinitely often. So the cofactor must not be split
> > into subfactors here.

>
> In a few more words: Maple simplifies (x-4) * 1/(x-4) -> 1; the second
> factor is not differentiable at x = 4, whereas the Maple-simplified
> product is. The validity (in Maple, as in most or all CAS) of this
> simplification is the source of the inconsistency: a non-differentiable
> cofactor of a Dirac delta can vanish, and a meaningless product
> involving a delta thereby become meaningful. Note that in a meaningless
> product associativity doesn't hold! Strictly speaking, a product
> involving a Dirac delta must be declared meaningless if any one cofactor
> cannot be differentiated an infinite number of times. I suppose this can
> relaxed by checking the product of all cofactors for differentiability
> after its simplification (in Maple) - provided that associativity
> involving the Dirac delta is never assumed to hold until such test is
> passed.
>

> >
> > The Dirac delta is a so-called "distribution"; these object are
> > defined via their action on certain spaces of "test functions". You
> > may for example refer to Constantinescu (1974), "Distributionen und
> > ihre Anwendungen in der Physik", but any other text on "distributions"
> > should do as well.

>

You misunderstood my question: I was asking about CAS so I am
asking about algebraic rules. And I used simple example to
ilustrate the problem, but real worry is when CAS is doing
some longish computation split into evaluationg several
expression. At any given time CAS sees only part of computation
and does not know if the result will in the future meet
delta functions. Normally CAS assumes that it deals
with field. If that is true then CAS is consistent with
itself. Schoolboy (or numeric) treatment of square roots
breaks field assumption, but there are papers which
propose a few treaks and prove that is specific context
this leads to correct results.

I do have few ideas to try, but I hoped that there is
some existing research.


--
Waldek Hebisch
hebisch@math.uni.wroc.pl



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