Posted by
Robert Howard-2-3 on
URL: http://friam.383.s1.nabble.com/Seminal-Papers-in-Complexity-tp524047p524114.html
Exactly! Linear functions (or operators) can be broken into smaller,
summable pieces where all terms can be scaled simultaneously.
You can factor common things out and operate on parts in a piecemeal manner;
e.g. the Taylor or Fourier Series.
When you get into non-linear stuff, you lose this fantastic mathematical
tool of manipulation and simplification; i.e. factoring out (or distributing
in) something common over a set of parts.
Linear equations often give you the ability to look at one part of the
equation in isolation from the rest; sort of like a spectral series.
It's easier to understand because you don't have to look at everything at
once.
That's why non-linear functions (like some integral equations) are more
often than not, very difficult to analyze, and often impossible to solve
analytically.
Our brains must think linearly. One can extend an imaginary line to any
distance with perfect accuracy. But curve it in a non-trivial way, and the
accuracy quickly attenuates.
Robert Howard
Phoenix, Arizona
-----Original Message-----
From:
[hidden email] [mailto:
[hidden email]] On Behalf
Of Russell Standish
Sent: Friday, June 22, 2007 12:52 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:
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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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