[FRIAM] another idea for a generalized "nonlinearity" (was Re: Seminal Papers in Complexity)

[Phil Henshaw]

Just to rephrase, there's a great way to reapply all the basic theorems
of calculus directly to real physical processes (skipping the
interceding equations).  Use data curves with an appropriate rule for
determining a value and slope at any point by iteration.  Works great
and provides a crystal clear identification of the emergent non-linear
phases of real processes.  

Like anything, you'd expect many questions, and slow beginning, then big
strides.   One of the hurdles is the software...  As powerful as they
are I hate R, and Excel, and AutoCad, though I have nothing else to
use... 


Phil Henshaw                       ¸¸¸¸.·´ ¯ `·.¸¸¸¸
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e-mail: pfh at synapse9.com          
explorations: www.synapse9.com    


> -----Original Message-----
> From: friam-bounces at redfish.com 
> [mailto:friam-bounces at redfish.com] On Behalf Of Glen E. P. Ropella
> Sent: Friday, June 22, 2007 3:02 PM
> To: The Friday Morning Applied Complexity Coffee Group
> Subject: [FRIAM] another idea for a generalized 
> "nonlinearity" (was Re: Seminal Papers in Complexity)
> 
> 
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> I just realized there's another general sense of "linearity" 
> that some non-mathematical descriptions target, that of 
> "balance".  The idea is that a system shows some sort of 
> balance where no one component contributes more than any 
> other component.  Simple examples would be adding a nonlinear 
> term to a previously linear equation:
> 
>    1) z = a*x + b*y, changed to
>    2) z = a*x^2 + b*y
> 
> Technically, (2) is linear because f(x,y) = f(x) + f(y) (note 
> that just because the sets described are not planes doesn't 
> mean the function is nonlinear).  It is still describable as 
> linear because one can cleanly separate out the co-domain (by 
> definition) into X and Y.  I.e. in the characterization of 
> the co-domain, X and Y contribute equally, any point in that 
> product space is fair game.
> 
> But, if we were to bias it in some way, let's say we define 
> functions as going from the positive reals (R+) crossed with 
> the reals (f : R+ x R -> R).  Then that may touch on 
> someone's intuition of what "nonlinear" means.
> 
> That sort of concept is captured in linear algebra by the 
> concept of a "balanced set".  E.g. R+ x R is not balanced 
> because R+ is not balanced.  The set described by (2) above 
> is not balanced where (1) above _is_ balanced, even though 
> both are linear functions.  Of course, in order for one to 
> have a sense of balance, one has to have a fulcrum about 
> which to balance.  And sometimes its useful to describe 
> spaces that don't have such fulcrums (as in the affine plane 
> described previously).  So the linear algebra "balanced set" 
> doesn't generalize very well, especially to vague 
> descriptions of spaces and mappings between them.
> 
> Glen E. P. Ropella wrote:
> > But, there's no reason you couldn't define the same _type_ of thing 
> > with other composition operators.  All you need to do to have an 
> > unambiguous definition of what you mean by "linearity" is 
> to a) define 
> > the composition operator you're talking about and b) define the 
> > closure of that operator.  Of course there are plenty of such 
> > constructs already, they just aren't referred to with the word 
> > "linearity".
> 
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> - --
> glen e. p. ropella, 971-219-3846, http://tempusdictum.com
> I have an existential map. It has 'You are here' written all 
> over it. -- Steven Wright
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> ============================================================
> 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
> 
>