Fred is right that many terms that might apply are overloaded. I like "pull-back" for this notion, and I think it's unambiguous: "the pull-back of G by F" is definitely x |-> F^-1(G(F(x))). On the other hand, people might look at you funny for using it. The motivation: think of F : X->Y and G : Y->Y. Then if you have a point x in X, you can make G act on it even though it lives in the wrong space, by x |-> F^-1(G(F(x))). So you are pulling the function G back through the map F to get a new function defined on X; this new function is the pull-back. So if you call this the pull-back in a situation where the space-change point of view is unnatural, you might get odd looks... --Michael Kleber On Feb 16, 2008 10:24 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
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.
That's what it's called in group theory of course; here the group operation is functional composition (at least for functions well-enough behaved).
The trouble with a lot of these terms is that they become overloaded with alternative meanings in other initially separate domains of mathematics, which in due course overlap and cause ambiguity.
In geometric algebra, the transform of a flat Y by an isometry X may be represented by X^{-1} Y X, the group-theory conjugate; however X^{-1} is essentially given by a component sign-change (reversion), which in the case of quaternion X is just the quaternion conjugate. To make things worse, composing this with the parity involution is then called the Clifford conjugate.
So we really could use an alternative nomenclature here, but I haven't come up with any so far. Maybe I shall have to settle for Henry's "lifting" (from differential geometry?) --- any more ideas, anybody?
WFL
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