Posted by
Russell Standish on
URL: http://friam.383.s1.nabble.com/Seminal-Papers-in-Complexity-tp524047p524111.html
On Fri, Jun 22, 2007 at 10:34:09AM -0700, Glen E. P. Ropella wrote:
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> 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.
--
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