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Re: comments re draft version 2.0
Stan Devitt writes:
> The (D^2)(y) --> D(y)*D(y) interpretation is
> exactly the interpretation you get in, for example, Maple.
> Maple has the somewhat unique feature of supporting an
> algebra of functions so that (f+g)(x) -> f(x) + g(x), etc.
> and (1)(x) -> 1 .
<aside>
In Maple (D^2)(sin) seems to require both pairs of parentheses
and the same result is available with less typing via the more
natural D(sin)^2 . The second derivative seems to require
(D@@2)(sin)
</aside>
Maple's D is a derivation acting on the algebra of functions.
And, yes, the default for multiplication in the algebra of functions
is point-wise multiplication. But D is not a member of the function
algebra.
> The real question is "what meaning" do wish to associate with
>
> \apply{D^2}{y}.
>
Then isn't the question: What does D^2 mean? (Certainly one does not
want to propose for LaTeX-like markup a command where the meaning of
an argument as an input, as opposed to the expanded result, depends on
the command and the other arguments.)
> I claim that the answer may depend on the properties of D, and
> that even then, there is more than one reasonable meaning - at least
> one based on operator composition, and one based on product.
This is correct for ring valued operators but not for vector valued
operators or set valued operators. In both of the latter cases the
composition D \circ D is defined for operators when range is a subset
of domain.
But now for TeX-like author markup which of the possible meanings will
an author take as an implied default?
> So long as the author can say which definition is to be used (and we
> can) , we can over-ride whatever default meaning is chosen.
> The outcome can depend on the signature, for example, the presence
> of an operator versus a symbol.
Yes, if it is meant that the author can say which meaning (in D^2 y)
for the "^" is to be used. But what, again, what does an author see
as the default?
For TeX-like markup if we want to bring authors along, then we need to
be close to traditional TeX. The behavior of a well-known computer
algebra system (that I happen to use) is less relevant than traditional
TeX markup practice.
I see no way to get authors to agree to use \compose{D}{N} y for
the N-th composition power given that they've been using D^N y since
the time of Newton and their readers have been understanding it since
then.
Now why do those (sophisticated) authors and readers succeed
with D^N y ?
Multiplicative operations on *functions* include, at the very least,
point-wise multiplication, composition, and various convolutions.
For *functions* authors see point-wise multiplication as the default
choice in this list. Other multiplications need explicit mention.
Multiplicative operations on *operators* are a *very* different
matter. The most basic one is composition.
Value-wise multiplication is subordinate to the multiplication that is
the default for values of the operator. Mathematical authors write
(D y)^2 for the value-wise square using the default meaning of "^" for
functions. And there is no very convenient LaTeX or TeX way for an
author to refer to the operator
y \mapsto (D y)^2 ,
nor do I see a serious need for one. An author would simply write
this display and give it a name.
I am not, however, suggesting change either for Maple or for MathML
on this point.
We do need tools for creating content MathML and, in particular, ways
of so doing that are as close as possible to LaTeX.
-- Bill
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