From: William F. Hammond <hammond@csc.albany.edu>

Date: Thu, 13 Apr 2000 13:28:54 -0400 (EDT)

Message-Id: <200004131728.NAA12710@hilbert.math.albany.edu>

To: jsdevitt@radicalflow.com

Cc: www-math@w3.org

Date: Thu, 13 Apr 2000 13:28:54 -0400 (EDT)

Message-Id: <200004131728.NAA12710@hilbert.math.albany.edu>

To: jsdevitt@radicalflow.com

Cc: www-math@w3.org

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. -- BillReceived on Thursday, 13 April 2000 13:29:32 UTC

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