"represent these things" seems further to imply that the correspondence goes both ways, or at least that the degree of such functions is unbounded. Is that the case? Where does one learn about such matters? WFL On 8/28/14, Adam P. Goucher <apgoucher@gmx.com> wrote:
It's infinite-dimensional.
We can certainly represent these things as maps of the form:
(x, y, z) --> (f(x,y,z), g(x,y,z), h(x,y,z)) [for two dimensions; generalise as appropriate]
where f, g and h are homogeneous rational functions, where deg(numerator) = deg(denominator) + 1.
Sent: Thursday, August 28, 2014 at 8:34 PM From: "Fred Lunnon" <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Stupid question about geometrical transformations
Oops --- should read (n+1)^2-1 + (n+2)(n+1)/2 - (n+1)n/2 - 1 = n^2 + 3n .
One immediate difficulty is incompatible compactifications --- a hyperplane versus a single point at infinity.
WFL
On 8/28/14, Fred Lunnon <fred.lunnon@gmail.com> wrote:
What is generated by the union of projective and conformal (Moebius) groups?
Since these two intersect in similarities, the super-group in n-space has dimension at least (n^2-1) + (n+2)(n+1)/2 - (n+1)n/2 - 1 = n^2 + n - 1 ; just how big is it?
How should such transformations be represented for computational purposes?
Why don't I know the answers to these apparently obvious questions? [Uh, maybe don't answer that one right now ...]
Physicists have previously devoted some thought to this matter: in particular, a promising paper by Wolfgang Bertram (2001) at http://www.emis.de/journals/AG/2-4/2_329.pdf launches into discussing "Jordan functors", which will however surely cost this innocent much gruesome effort to decode.
[Pascual Jordan certainly seems put himself about, despite which I don't recall ever having encountered him before this week.]
Fred Lunnon
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