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>

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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