On Fri, Jun 22, 2007 at 10:34:09AM -0700, Glen E. P. Ropella wrote: > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > Michael Agar wrote: > > As described in past posts, that's exactly what I'm trying to figure > > out--formal math definition doesn't help, metaphorical use too vague. > > Whatever the solution is, it's likely to be propositional/schematic > > rather than numeric and involve observer perspective/background > > knowledge. I'll write more to the list when I think I'm onto a solution. > > Formal math definitions do help. You just can't be myopic about it and > restrict yourself to arithmetic. Open it up to higher math. > > It seems you want to generalize linearity to apply to _other_ > composition functions. The typical definition of linearity applies only > to addition, i.e. f(x+y) != f(x) + f(y). If you abstract up just a bit, > linearity means "on the same line", which is a way of saying "in the > same space" where the space is 1 dimensional. It's simply a closure > under addition.
Not just addition, but also scalar multiplication by a member of a field. For any group G, one can consider the class of functions f:G->G satisfying f(x+y)=f(x)+f(y). This induces a linear-like property over N x G, ie for all a, b in N and for all x and y in G, f(ax+by) = af(x)+bf(y) where ax = \sum_i=0^a x However such objects are not linear functions, and don't appear to have a name. Perhaps they're not all that useful. -- ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
