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Re: comments re draft version 2.0

From: Stan Devitt <jsdevitt@radicalflow.com>
Date: Thu, 13 Apr 2000 13:13:30 -0400
Message-ID: <002401bfa56b$973ffca0$6561a8c0@devitt.local>
To: "William F. Hammond" <hammond@csc.albany.edu>
Cc: <www-math@w3.org>
You correctly observe that readers have been successfully interpreting

    D^2 y   --> (D@D) (y)

and

    x^2 y   -->  x * x * y

(or for that matter   sin x  --> sin(x)   versus     a x   --> a * x )
essentially forever.  The information used to achieve this understanding
is derived from the context (subject matter of  paper, course )
and perhaps additional knowledge about
the roles and properties of  specific instances of the variables
communicated by adopting "standard" usage in a particular exposition.

But, software almost always encounters such expressions with less
context than the human reader, from the outset -- often just the expression
its-self.  Thus, in order for  software to successfully catch the
distinction the
encoding of the must carry that extra information (either explicitly or
through documented defaults)

(aside:  this double use of notation is a difficult point for many weaker
students as well,  perhaps because they  too have too little context.)




----- Original Message -----
From: William F. Hammond <hammond@csc.albany.edu>
To: <jsdevitt@radicalflow.com>
Cc: <www-math@w3.org>
Sent: Thursday, April 13, 2000 1:28 PM
Subject: 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
>
Received on Thursday, 13 April 2000 14:11:19 GMT

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