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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". ============================================================ 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
