Yes: the defining property (at any rate in 2-space) is that every loop can be shrunk to a point, while remaining within the region. Wikipedia is a good source for cribbing this kind of elementary stuff: even if you don't follow the (often terse) text, the references provide a useful starting point for further searches. But I'm afraid my notion still doesn't stand up: the surface might project up like a cigar at an angle from a loop inside R so that the tip lay outside it, and the test falls over again (along with the cigar). It looks as if I have to return to an earlier, more pedestrian technique, where the region is bounded by line segments orthogonal to the axes, and I reason about the 2-D critical points of x(y) and y(x) simultaneously. (I have become well confused through mixing abcissae and 2 and 3-D, resuting in elementary errors: sorry, everybody!) WFL On 5/29/14, David Makin <makinmagic@tiscali.co.uk> wrote:
Is "simply connected" topologically the same as a solid circle ? (I never formally studied topology)
On 29 May 2014, at 21:40, Fred Lunnon wrote:
Despite trying hard, I managed to foul up my proposed criterion --- what I actually had in mind was "surface z = f(x, y) has no critical points within R " --- but Warren has apparently discerned what I meant, and skewered it.
But maybe things can be patched up by specifying that R is simply connected?
[ " Dear Mr ... Thankyou for your proof of Fred's last theorem. The first mistake is on line ... of page ... " ]
WFL
On 5/29/14, Warren D Smith <warren.wds@gmail.com> wrote:
Lunnon: Given a plane curve C defined by f = 0 with f(x, y) polynomial, and a region R of the plane (possibly extending to infinity), I assert that C avoids R provided C avoids the boundary of R ; and C has no critical points within R .
There must be a well-known theorem to this effect (unless, of course, it's actually false...
WDS: a counterexample: C=unit circle, R = thickened version of unit circle (annulus).
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun