Is this true?
in [Math]
|
Prev: WSUrinal "cap & tax" terminology
Next: P= /V:P == 2P*V because fist '/V' is actually '/' divided by 'V'comp.language.python.brood of vipers! Look! I got it, Gus! I got it! Look: P= /V:P == 2P*V because fist '/V' is actually '/' divided by 'V'
From: cwldoc on 23 Jun 2010 13:13 > mike3 <mike4ty4(a)yahoo.com> writes: > > > Hi. > > > > Is it true that if one has a continuous curve > connecting two points, > > makes a copy of it and slides the copy along the > direction between > > those points a distance less than 1/2 the distance > between the points, > > then the original curve and translated copy will > intersect at at least > > one point? If not, can you provide me a > counterexample, and if so, how > > about either a proof or just a hint at the proof, > if the proof isn't > > too heavily sophisticated? > > Counterexample: points are (0,0) and (1,1), curve is > y = sin(9/2 pi x) - x > for 0 <= x <= 1, slide right by distance 0.45. > -- > Robert Israel > israel(a)math.MyUniversitysInitials.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, > BC, Canada It seems to work (the points being (0,0) and (1,0)), but the distance shifted is not 1/2 of the distance between the points. The difficulty for me in constructing a counterexample arises when you shift the curve exactly 1/2 the distance.
From: Bart Goddard on 23 Jun 2010 19:58 cwldoc <cwldoc(a)aol.com> wrote in news:1439632397.17084.1277326908185.JavaMail.root(a)gallium.mathforum.org: > The "original function" must actually be of the form: > f:R->R^2, where f is continuous, I don't think that's what the OP meant. The two points connected were (a,f(a)) and (b,f(b)). -- Cheerfully resisting change since 1959.
From: cwldoc on 24 Jun 2010 11:56 > cwldoc <cwldoc(a)aol.com> wrote in > news:1439632397.17084.1277326908185.JavaMail.root(a)gall > ium.mathforum.org: > > > > The "original function" must actually be of the > form: > > f:R->R^2, where f is continuous, > > I don't think that's what the OP meant. The two > points connected were (a,f(a)) and (b,f(b)). I don't see anything in the original post about (a,f(a)) and (b,f(b)). But anyway it's a moot point in light of the counterexamples offered by Robert Israel. > > > -- > Cheerfully resisting change since 1959.
From: Bart Goddard on 24 Jun 2010 16:07 cwldoc <cwldoc(a)aol.com> wrote in news:1335339125.22275.1277409416655.JavaMail.root(a)gallium.mathforum.org: >> cwldoc <cwldoc(a)aol.com> wrote in >> news:1439632397.17084.1277326908185.JavaMail.root(a)gall >> ium.mathforum.org: >> >> >> > The "original function" must actually be of the >> form: >> > f:R->R^2, where f is continuous, >> >> I don't think that's what the OP meant. The two >> points connected were (a,f(a)) and (b,f(b)). > > I don't see anything in the original post about (a,f(a)) and (b,f(b)). > But anyway it's a moot point in light of the counterexamples offered > by Robert Israel. Then note that the Robert's counterexample is a function from R to R, not R^2. -- Cheerfully resisting change since 1959.
From: cwldoc on 25 Jun 2010 10:22
> cwldoc <cwldoc(a)aol.com> wrote in > news:1335339125.22275.1277409416655.JavaMail.root(a)gall > ium.mathforum.org: > > >> cwldoc <cwldoc(a)aol.com> wrote in > >> > news:1439632397.17084.1277326908185.JavaMail.root(a)gall > >> ium.mathforum.org: > >> > >> > >> > The "original function" must actually be of the > >> form: > >> > f:R->R^2, where f is continuous, > >> > >> I don't think that's what the OP meant. The two > >> points connected were (a,f(a)) and (b,f(b)). > > > > I don't see anything in the original post about > (a,f(a)) and (b,f(b)). > > But anyway it's a moot point in light of the > counterexamples offered > > by Robert Israel. > > Then note that the Robert's counterexample is a > function > from R to R, not R^2. That is correct. The curves represented by continuous functions from R to R are a subset of the continuous curves in the plane. However, some curves cannot be represented in that way, for example a spiral. The counterexample given disproved the original assertion. However if the assertion had been true, then it would not have been sufficient to prove it for only the subset of cases representable by functons from R to R. > > -- > Cheerfully resisting change since 1959. |