Seminal Papers in Complexity

Posted by Phil Henshaw-2 on
URL: http://friam.383.s1.nabble.com/Seminal-Papers-in-Complexity-tp524047p524115.html

Na, I think even the most sophisticated math misses all the truly supple
shape of natural form, and it it's of huge signifiance in our
missunderstanding of natural phenomena.


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> -----Original Message-----
> From: friam-bounces at redfish.com
> [mailto:friam-bounces at redfish.com] On Behalf Of Russell Standish
> Sent: Friday, June 22, 2007 3:52 PM
> To: The Friday Morning Applied Complexity Coffee Group
> Subject: Re: [FRIAM] Seminal Papers in Complexity
>
>
> 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.
>
>
> --
>
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> --------------
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