[math-fun] curves in regions
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).
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
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.
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
Was the region R meant to be restricted to a single continuous area without holes (i.e. topologically a solid circle but maybe extending to infinity) ? On 29 May 2014, at 20:17, Warren D Smith 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
The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
Suppose f(x,y) = y, so C is the x-axis, there are no critical points, and R is given by (x,y) is in R when |y| < 1. Maybe if R is compact and simply connected? On Thu, May 29, 2014 at 4:46 PM, David Makin <makinmagic@tiscali.co.uk> wrote:
Was the region R meant to be restricted to a single continuous area without holes (i.e. topologically a solid circle but maybe extending to infinity) ?
On 29 May 2014, at 20:17, Warren D Smith 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
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
-- Andy.Latto@pobox.com
My vague memory from algebraic geometry in college is that theorems of this sort start being true more reliably in *complex, projective* space. R being compact and simply-connected is still skewered by Warren's example, replacing the annulus with a solid disk. But in complex space, the circle has parts that run out to infinity, so it cuts the disk boundary somewhere. On Thu, May 29, 2014 at 7:15 PM, Andy Latto <andy.latto@pobox.com> wrote:
Suppose f(x,y) = y, so C is the x-axis, there are no critical points, and R is given by
(x,y) is in R when |y| < 1.
Maybe if R is compact and simply connected?
On Thu, May 29, 2014 at 4:46 PM, David Makin <makinmagic@tiscali.co.uk> wrote:
Was the region R meant to be restricted to a single continuous area without holes (i.e. topologically a solid circle but maybe extending to infinity) ?
On 29 May 2014, at 20:17, Warren D Smith 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
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
-- Andy.Latto@pobox.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Suppose the function is f(x, y) = x^2 + y^2 ; then the maximum over the origin projects into the unit disc, so the 3-space method correctly rejects it as a zero-free region. WFL On 5/30/14, Allan Wechsler <acwacw@gmail.com> wrote:
My vague memory from algebraic geometry in college is that theorems of this sort start being true more reliably in *complex, projective* space. R being compact and simply-connected is still skewered by Warren's example, replacing the annulus with a solid disk. But in complex space, the circle has parts that run out to infinity, so it cuts the disk boundary somewhere.
On Thu, May 29, 2014 at 7:15 PM, Andy Latto <andy.latto@pobox.com> wrote:
Suppose f(x,y) = y, so C is the x-axis, there are no critical points, and R is given by
(x,y) is in R when |y| < 1.
Maybe if R is compact and simply connected?
On Thu, May 29, 2014 at 4:46 PM, David Makin <makinmagic@tiscali.co.uk> wrote:
Was the region R meant to be restricted to a single continuous area without holes (i.e. topologically a solid circle but maybe extending to infinity) ?
On 29 May 2014, at 20:17, Warren D Smith 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
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
-- Andy.Latto@pobox.com
_______________________________________________ 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
participants (5)
-
Allan Wechsler -
Andy Latto -
David Makin -
Fred Lunnon -
Warren D Smith