Is there a common name for the idiom F^-1( G( F( x ) ) ) ? Examples Inverse_FFT( G( FFT( signal ) ) ) translate( x, rotate( a, translate( -x, picture ) ) ) (rotate around (x,0) ) Twist top of Rubik's cube CW, side CW top CCW. Open drawer remove silverware Close drawer Move smallest disk from A to C Move bigger disk from A to B Move smallest disk from C to A --Steve
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. /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--
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
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.
On Feb 16, 2008 7:46 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
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...
Category theorists always get odd looks... *sigh* -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
"lifting" ? performing an operation in a transform domain ? walk a mile in someone (else)'s shoes ? At 03:58 PM 2/16/2008, Steve Witham wrote:
Is there a common name for the idiom F^-1( G( F( x ) ) ) ?
Examples Inverse_FFT( G( FFT( signal ) ) )
translate( x, rotate( a, translate( -x, picture ) ) ) (rotate around (x,0) )
Twist top of Rubik's cube CW, side CW top CCW.
Open drawer remove silverware Close drawer
Move smallest disk from A to C Move bigger disk from A to B Move smallest disk from C to A
--Steve
participants (6)
-
Bernie Cosell -
Fred lunnon -
Henry Baker -
Michael Kleber -
Mike Stay -
Steve Witham