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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