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Topic: Stone Cech
Replies: 49   Last Post: Mar 28, 2013 12:15 PM

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: Stone Cech
Posted: Mar 22, 2013 10:43 AM
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On Fri, 22 Mar 2013 05:34:37 -0500, quasi <quasi@null.set> wrote:

>Butch Malahide wrote:
>>Butch Malahide wrote:
>>>Butch Malahide wrote:
>>>>quasi wrote:
>>>>>Butch Malahide wrote:
>>>>>>quasi wrote:
>>>>>>>quasi wrote:
>>>>>>>>quasi wrote:
>>>>>>>>>quasi wrote:
>>>>>>>>>>Butch Malahide wrote:
>>>>>>>>>>>David C. Ullrich wrote:
>>>>>>>>>>>>Butch Malahide wrote
>>>>>>>>>>>>>William Elliot wrote:
>>>>>>>>>>>>>>David Hartley wrote:
>>>>>>>>>>>>>>>William Elliot wrote:
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>Perhaps you could illustrate with the five
>>>>>>>>>>>>>>>>different one to four point point
>>>>>>>>>>>>>>>>compactifications of two open end line
>>>>>>>>>>>>>>>>segements.

>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>(There are seven.)

>>>>>>>>>>>>>>
>>>>>>>>>>>>>>Ok, seven non-homeomophic finite Hausdorff
>>>>>>>>>>>>>>compactications.

>>>>>>>>>>>>>
>>>>>>>>>>>>>How many will there be if you start with n segments
>>>>>>>>>>>>>instead of 2?

>>>>>>>>>>>>
>>>>>>>>>>>>Surely there's no simple formula for that?
>>>>>>>>>>>>
>>>>>>>>>>>> ...

>>>>>>>>>>>
>>>>>>>>>>> ...
>>>>>>>>>>>
>>>>>>>>>>>I wasn't necessarily expecting a *complete* answer,
>>>>>>>>>>>such as an explicit generating function. Maybe someone
>>>>>>>>>>>could give a partial answer, such as an asymptotic
>>>>>>>>>>>formula, or nontrivial upper and lower bounds, or a
>>>>>>>>>>>reference to a table of small values, or the ID number
>>>>>>>>>>>in the Encyclopedia of Integer Sequences, or just the
>>>>>>>>>>>value for n = 3. (I got 21 from a hurried hand count.)

>>>>>>>>>>
>>>>>>>>>>For n = 3, my hand count yields 19 distinct
>>>>>>>>>>compactifications, up to homeomorphism.
>>>>>>>>>>
>>>>>>>>>>Perhaps I missed some cases.
>>>>>>>>>>

>>>>>>>>>I found 1 more case.
>>>>>>>>>
>>>>>>>>>My count is now 20.

>>>>>>>>
>>>>>>>>I found still 1 more case.
>>>>>>>>
>>>>>>>>So 21 it is!
>>>>>>>>
>>>>>>>>But after that, there are no more -- I'm certain.

>>>>>>>
>>>>>>>Oops -- the last one I found was bogus.
>>>>>>>
>>>>>>>So my count is back to 20.

>>>>>>
>>>>>>Hmm. I counted them again, and I still get 21.
>>>>>>
>>>>>>4 3-component spaces: OOO, OO|, O||, |||.
>>>>>>
>>>>>>7 2-component spaces: OO, O|, O6, O8, ||, |6, |8.
>>>>>>
>>>>>>10 connected spaces: O, |, 6, 8, Y, theta, dumbbell, and
>>>>>>the spaces obtained by taking a Y and gluing one, two, or
>>>>>>all three of the endpoints to the central node.

>>>>>
>>>>>Thanks.
>>>>>
>>>>>It appears I missed the plain "Y", but other than that,
>>>>>everything matches.
>>>>>
>>>>>So yes, 21 distinct types.

>>>>
>>>> For n = 4 I get 56 types. If I counted right (very iffy),

>>>
>>> Found two more. Never mind!

>>
>>And now I get 61. The hell with it.

>
>Ullrich predicted it (hopeless squared).


Hey, you guys are making progress!
Now all we need is an f such that
f(2) = 7, f(3) = 21 or whatever it was,
and f(4) = 61. I doubt there's more than
one such f...

>
>For small n, say n < 10, it might be feasible to get the counts
>via a computer program, but my sense is that the development of
>such a program would be fairly challenging. If I get a chance,
>I may give it a try.


My prediiction is that anything remotely resembling brute
force will take forever to run, even in a compiled language.
Just enumerating all the possible equivalence relations
on 2n points is going to take a long time; now a
brute force check whether _one_ pair of equivalence
relations defines homeomorphic spaces will also take
a long time, and there will be N(N-1)/2 such checks,
where N is huge even for small n.

But don't let me stop you. I'd start in Python, which
would be much too slow - the point being that Python
is so easy to _write_ that it would be the quickest
way to get the algorithm straight. I'd test the
Python code for n = 2 and 3 maybe; for n = 3
it would take forever. Once I had a correct
algorithm I'd rewrite it in a compiled language.
Then ask for the answer for n = 4 and maybe
take a trip somewhere while waiting for the
answer...

>
>quasi




Date Subject Author
3/14/13
Read Stone Cech
William Elliot
3/14/13
Read Re: Stone Cech
fom
3/15/13
Read Re: Stone Cech
fom
3/16/13
Read Re: Stone Cech
William Elliot
3/15/13
Read Re: Stone Cech
David C. Ullrich
3/17/13
Read Re: Stone Cech
William Elliot
3/17/13
Read Re: Stone Cech
David C. Ullrich
3/17/13
Read Re: Stone Cech
fom
3/18/13
Read Re: Stone Cech
David C. Ullrich
3/18/13
Read Re: Stone Cech
fom
3/18/13
Read Re: Stone Cech
David Hartley
3/19/13
Read Re: Stone Cech
William Elliot
3/19/13
Read Re: Stone Cech
David Hartley
3/19/13
Read Stone Cech
William Elliot
3/20/13
Read Re: Stone Cech
Butch Malahide
3/20/13
Read Re: Stone Cech
David C. Ullrich
3/20/13
Read Re: Stone Cech
Butch Malahide
3/20/13
Read Re: Stone Cech
Butch Malahide
3/21/13
Read Re: Stone Cech
quasi
3/21/13
Read Re: Stone Cech
quasi
3/21/13
Read Re: Stone Cech
quasi
3/21/13
Read Re: Stone Cech
quasi
3/21/13
Read Re: Stone Cech
Butch Malahide
3/21/13
Read Re: Stone Cech
quasi
3/22/13
Read Re: Stone Cech
Butch Malahide
3/22/13
Read Re: Stone Cech
Butch Malahide
3/22/13
Read Re: Stone Cech
Butch Malahide
3/22/13
Read Re: Stone Cech
quasi
3/22/13
Read Re: Stone Cech
David C. Ullrich
3/22/13
Read Re: Stone Cech
David C. Ullrich
3/22/13
Read Re: Stone Cech
Butch Malahide
3/23/13
Read Re: Stone Cech
Butch Malahide
3/23/13
Read Re: Stone Cech
David C. Ullrich
3/23/13
Read Re: Stone Cech
David C. Ullrich
3/23/13
Read Re: Stone Cech
Frederick Williams
3/23/13
Read Re: Stone Cech
David C. Ullrich
3/23/13
Read Re: Stone Cech
Frederick Williams
3/22/13
Read Re: Stone Cech
Butch Malahide
3/23/13
Read Re: Stone Cech
David C. Ullrich
3/22/13
Read Re: Stone Cech
Butch Malahide
3/23/13
Read Re: Stone Cech
quasi
3/23/13
Read Re: Stone Cech
Butch Malahide
3/23/13
Read Re: Stone Cech
Butch Malahide
3/24/13
Read Re: Stone Cech
quasi
3/24/13
Read Re: Stone Cech
Frederick Williams
3/24/13
Read Re: Stone Cech
quasi
3/25/13
Read Re: Stone Cech
Frederick Williams
3/28/13
Read Re: Stone Cech
Frederick Williams
3/25/13
Read Re: Stone Cech
quasi
3/19/13
Read Re: Stone Cech
David C. Ullrich

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