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From: David C. Ullrich on 28 May 2010 07:26
On Thu, 27 May 2010 12:24:53 -0700 (PDT), Ray Vickson
<RGVickson(a)shaw.ca> wrote:

>On May 27, 11:19�am, "Dave L. Renfro" <renfr...(a)cmich.edu> 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
>
>This reminds me of a problem I posted to this group several years ago,
>and did not receive a convincing answer. Take [0,1]^2 with a uniform
>distribution of mass whose total weight = 1. Cut out a Lebsegue non-
>measurable set in [0,1]^2. How much does it weigh?

The question is meaningless. Weight is a property of physical
objects, and that set is not a physical object.

I imagine that's an example of an answer that's not "convincing",
but it's true regardless.

>
>R.G. Vickson

From: William Hughes on 28 May 2010 07:30
On May 27, 4:24 pm, Ray Vickson <RGVick...(a)shaw.ca> wrote:
> On May 27, 11:19 am, "Dave L. Renfro" <renfr...(a)cmich.edu> 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
>
> This reminds me of a problem I posted to this group several years ago,
> and did not receive a convincing answer. Take [0,1]^2 with a uniform
> distribution of mass whose total weight = 1. Cut out a Lebsegue non-
> measurable set in [0,1]^2. How much does it weigh?
>
> R.G. Vickson

If the set, call it
A, is not measurable, then it does not have a weight, [There is
a sense in which weight(A) <= 1.]

- William Hughes
From: Dave L. Renfro on 28 May 2010 09:20
Rob Johnson wrote (in part):

> Looking back at the thread, I see that Robert Israel has
> come up with the same answer using a probabilistic argument.

Wow, two solutions between the time I left (Thur. afternoon)
and the time I returned (Fri. morning)!

For the record, here's one place (the only place?) the problem
can be found in the published literature:

Problem 116 (proposed by J. Wilker), Solution (by M. Shiffman and
S. Spital), Canadian Mathematical Bulletin 11 #2 (1968), 306-307.
http://books.google.com/books?id=mEpsils2t-0C&pg=PA306&lpg=PA306

The solution they arrive at is

-cos(2*pi/3)*cos(2*pi/3^2)*cos(2*pi/3^3)* ... *cos(2*pi/3^n)* ...

= 0.37143736...

Dave L. Renfro
From: Pubkeybreaker on 28 May 2010 10:06
On May 28, 3:45 am, Robert Israel
<isr...(a)math.MyUniversitysInitials.ca> wrote:
> "Dave L. Renfro" <renfr...(a)cmich.edu> writes:
>

>
> It's most convenient to consider the circular Cantor set as the image
> of the usual Cantor set under the map f: t -> exp(2 pi i t) from [0,1]
> into the complex plane.  Now if X_j are independent Bernoulli random
> variables with p=1/2 (i.e. flips of a fair coin),
> Y = sum_{j=1}^infty (2/3^j) X_j is uniformly distributed on
> the Cantor set.

I do not see where Y comes from. Can you explain further?
Excuse my ignorance.


> Thus you want
>
> E[exp(2 pi i Y)] = product_{j=1}^infty E[exp((4/3^j) pi i X_j)]
>                  = product_{j=1}^infty (1 + exp(4 pi i/3^j))/2
>                  = - product_{j=1}^infty cos(2 pi/3^j)

Yes. This is clear.


From: Pubkeybreaker on 28 May 2010 10:12
On May 28, 6:40 am, r...(a)trash.whim.org (Rob Johnson) wrote:
> In article <2de3769a-6d5b-4c72-9e1f-43e877f41...(a)c11g2000vbe.googlegroups..com>,

>
> Looking back at the thread, I see that Robert Israel has come up
> with the same answer using a probabilistic argument.
>
> Rob Johnson <r...(a)trash.whim.org>
> take out the trash before replying
> to view any ASCII art, display article in a monospaced font

Your posts always impress me.
Did you write: JUST THE ESSENTIALS OF ELEMENTARY STATISTICS?
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