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Re: Limit and Root Objects
*To*: mathgroup at smc.vnet.net
*Subject*: [mg72686] Re: Limit and Root Objects
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Mon, 15 Jan 2007 05:23:31 -0500 (EST)
*Organization*: The University of Western Australia
*References*: <NDBBJGNHKLMPLILOIPPOKEDJFFAA.djmp@earthlink.net> <eocs96$6s7$1@smc.vnet.net>
In article <eocs96$6s7$1 at smc.vnet.net>,
Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> Remember the point is that you cannot define such
> a function that will be continuous over the entire 6 dimensional (3
> complex dimensions) space of polynomials of degree 3 with complex
> roots (we actually normally remove the discriminant from the space).
But can you give a concrete example? I am interested in knowing the
locations/structure of the discontinuities for a specific simple
polynomial of degree 3 with complex coefficients.
> You are using a smaller subspace ( you have just one complex
> parameter) and over a smaller subspace naturaly it is possible to
> have a continuous root.
> If you look carefully at Adam's argument you will be easily able to
> see what must go wrong over the entire space of complex cubics.
Actually, no. I think that I get the essential idea but I am still not
able to see what must go wrong over the entire space of complex cubics.
A concrete example would help.
> This in fact is a good illustration of what a double edged weapon
> graphics and animations are in studying mathematics: they can just as
> easily mislead your intuition and lead you to wrong conclusions as to
> right ones. Which is one reason why I think one should never rely too
> much on such tools when teaching mathematics. Proofs are proofs and
> no number of "convincing animations" and "experimental mathematics"
> can replace them.
But if you can show an animation where the discontinuity is apparent
then you would convince me! And I would then try harder to better
understand Adam's argument.
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
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