center of mass of the Cantor set
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From: David R Tribble on 29 May 2010 11:56 Pubkeybreaker wrote: >> Did you write: JUST THE ESSENTIALS OF ELEMENTARY STATISTICS? > Rob Johnson wrote: > It wasn't me. Nor did I sell my soul at the crossroads to play the > blues (enter "sell my soul at the crossroads to play the blues" into > Google). But of course you still enjoy the blues, yes? "The Blues ain't nothin' but a good man feelin' bad."
From: Rob Johnson on 29 May 2010 20:50 In article <d4839f35-d5a5-4ff9-9f61-4d812dadeee4(a)e6g2000vbm.googlegroups.com>, David R Tribble <david(a)tribble.com> wrote: >Pubkeybreaker wrote: >>> Did you write: JUST THE ESSENTIALS OF ELEMENTARY STATISTICS? >> > >Rob Johnson wrote: >> It wasn't me. Nor did I sell my soul at the crossroads to play the >> blues (enter "sell my soul at the crossroads to play the blues" into >> Google). > >But of course you still enjoy the blues, yes? I would say that I have a fond appreciation of the blues. Rob Johnson <rob(a)trash.whim.org> take out the trash before replying Who put the tribbles in the quadrotriticale?
From: Robert Israel on 30 May 2010 04:10 Niels Diepeveen <n659474(a)dv1.demon.nl> writes: > Dave L. Renfro wrote: > > > Here's an interesting problem I recently saw. > > > > Take the usual middle thirds Cantor set, constructed on the > > closed interval [0, 2*pi] instead of the closed interval > > [0,1], and bend it without stretching into a circle of > > radius 1 centered at the origin of the xy-coordinate plane > > so that the points 0 and 2*pi are glued together at (0,1). > > What are the xy-coordinates for the center of mass of the > > resulting circular Cantor set, assuming a uniform density > > for the Cantor set? > > > > Dave L. Renfro > > I'm curious. When I first saw this post, I imagined the answer was > "Right next to the centre of mass of the rationals". > Usually, calculations of centres of mass are based on counting measure > for finite sets or Lebesue measure for infinite sets. Neither applies in > this case, yet everyone who replied seemed to make to make > (essentially) the same assumptions. Is there some general definition > that I'm not aware of, or is it really an ad hoc solution based on "it > stands to reason" given the symmetries. The Cantor set has a "natural" measure, which corresponds to Haar measure on {0,1}^N (N = natural numbers), and this is pretty clearly what was meant, although "uniform density for the Cantor set" is not good terminology. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Niels Diepeveen on 30 May 2010 07:36 Robert Israel wrote: > Niels Diepeveen <n659474(a)dv1.demon.nl> writes: > >> Dave L. Renfro wrote: >> >> > Here's an interesting problem I recently saw. >> > >> > Take the usual middle thirds Cantor set, constructed on the >> > closed interval [0, 2*pi] instead of the closed interval >> > [0,1], and bend it without stretching into a circle of >> > radius 1 centered at the origin of the xy-coordinate plane >> > so that the points 0 and 2*pi are glued together at (0,1). >> > What are the xy-coordinates for the center of mass of the >> > resulting circular Cantor set, assuming a uniform density >> > for the Cantor set? >> > >> > Dave L. Renfro >> >> I'm curious. When I first saw this post, I imagined the answer was >> "Right next to the centre of mass of the rationals". >> Usually, calculations of centres of mass are based on counting measure >> for finite sets or Lebesue measure for infinite sets. Neither applies in >> this case, yet everyone who replied seemed to make to make >> (essentially) the same assumptions. Is there some general definition >> that I'm not aware of, or is it really an ad hoc solution based on "it >> stands to reason" given the symmetries. > > The Cantor set has a "natural" measure, which corresponds to Haar measure > on {0,1}^N (N = natural numbers), and this is pretty clearly what was meant, > although "uniform density for the Cantor set" is not good terminology. I had heard about Haar measure, but I hadn't realized that it could be applied to this. Some more reading to do. It sounds like the sort of justification I was looking for. Thanks for the pointer. -- Niels Diepeveen
From: Niels Diepeveen on 30 May 2010 10:41 Rob Johnson wrote: > In article <4c008003$0$22920$e4fe514c(a)news.xs4all.nl>, > Niels Diepeveen <n659474(a)dv1.demon.nl> wrote: >>Dave L. Renfro wrote: >> >>> Here's an interesting problem I recently saw. >>> >>> Take the usual middle thirds Cantor set, constructed on the >>> closed interval [0, 2*pi] instead of the closed interval >>> [0,1], and bend it without stretching into a circle of >>> radius 1 centered at the origin of the xy-coordinate plane >>> so that the points 0 and 2*pi are glued together at (0,1). >>> What are the xy-coordinates for the center of mass of the >>> resulting circular Cantor set, assuming a uniform density >>> for the Cantor set? >>> >>> Dave L. Renfro >> >>I'm curious. When I first saw this post, I imagined the answer was >>"Right next to the centre of mass of the rationals". >>Usually, calculations of centres of mass are based on counting measure >>for finite sets or Lebesue measure for infinite sets. Neither applies in >>this case, yet everyone who replied seemed to make to make >>(essentially) the same assumptions. Is there some general definition >>that I'm not aware of, or is it really an ad hoc solution based on "it >>stands to reason" given the symmetries. > > There are several ways to characterize the uniform measure supported > on the Cantor set. One is defined in parallel to the way the Cantor > set is defined. For each stage of the middle thirds set, define the > measure to be the usual uniform measure on the remaining closed > intervals divided by their total measure. For example, > > u_0 = X_[0,1] > > u_1 = 3/2 X_{[0,1/3]U[2/3,1]} > > u_2 = (3/2)^2 X_{[0,1/9]U[2/91/3]U[2/3,7/9]U[8/9,1]} > > etc. > > Each measure is absolutely continuous and as measures, they converge > to the Cantor measure. I had dismissed this approach earlier, because I didn't see how it could be generalized in a meaningful way. Now that you made me look at it again, I'm starting to believe that it generalizes neatly to any compact inifinite set of reals. -- Niels Diepeveen
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