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 ¸¸¸¸.·´ ¯ `·.¸¸¸¸ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 680 Ft. Washington Ave NY NY 10040 tel: 212-795-4844 e-mail: [EMAIL PROTECTED] explorations: www.synapse9.com > -----Original Message----- > From: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] 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) > > > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > > 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". > > ============================================================ > 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 > > > > - -- > 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 > > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v1.4.6 (GNU/Linux) > Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org > > iD8DBQFGfBywZeB+vOTnLkoRAnM1AKDdMkLIf3LNW9pnhVA1M6wcoMQPMQCdERKI > UthB//12Jk4flYLe0c+PJhU= > =1Gja > -----END PGP SIGNATURE----- > > ============================================================ > 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 > > ============================================================ 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
