On 2/17/08, Bernie Cosell <bernie@fantasyfarm.com> wrote:
On 16 Feb 2008 at 18:58, Steve Witham wrote:
Is there a common name for the idiom F^-1( G( F( x ) ) ) ?
I think that was called "conjugation by F" when I ran across it.
From: Dan Asimov <dasimov@earthlink.net> What I know as the totally standard term for F^(-1) o G o F is the "conjugate" of G by F.
Don't you mean F o G o F^(-1)?
It has equivalent interpretations as
1) re-expressing G in terms of a new coordinate system as defined by F, and
2) performing the transformation G in "a new place" as defined by F. (E.g., if G is a rotation about the origin of R^2 about the origina, and F is a translation F(v) = v - v_0, then F^(-1) o G o F expresses the equivalent rotation of R^2 about v_0.
"In a new place" seems a more general metaphor than "in a new coordinate system" to me. Sometimes it feels more like "reach in". "Rotate the top of the cube, then the side, then rotate the top back." Doesn't easily map to ...that mapping for me. It's more like, do G with F out of the way...except there is some interference you will have to clean up. Another example, in a functional language, deconstruct a complicated structure, change one element, put the structure back together with the changed element. You can think of the slot that was changed as a plane of the space of the structure, but... I don't. "Incremental change" seems to be a common theme. Also, "look at it from a different point of view," which is ironic: For instance, as I understand it, a Fourier transform is a rotation in a Hilbert space?? So if I FT, then multiply by the frequency response curve I want, then IFT, what I did was "see it in frequency terms," which we metaphorically call "another point of view," but in the Hilbert space it really *is*...
From: James Propp <jpropp@cs.uml.edu>
And then Melzak went ahead and did just that, in a book called "Bypasses: A Simple Approach to Complexity".
From: Eugene Salamin <gene_salamin@yahoo.com> They're wonderful books. And you have no excuse for not reading them yourself, since they have been reprinted by Dover, "two volumes bound as one". http://store.doverpublications.com/0486457818.html
Neither that web site or Amazon mentions "Bypasses"--are you sure it's the second volume of that set? --Steve