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Topic: Another AC anomaly?
Replies: 280   Last Post: Dec 27, 2009 3:53 PM

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Daryl McCullough

Posts: 2,871
Registered: 1/29/05
Re: Another AC anomaly?
Posted: Nov 26, 2009 10:04 AM
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Bill Taylor says...

>One of my favourites is this; for naturals n:- compare...
>
>a) At each time 1 - 1/n, add balls numbered
> 2^(n-1) to 2^n - 1 to the pot, and remove ball number n.
>
>b) At each time 1 - 1/n, add 2^(n-1) - 1 balls to the pot, and
> replace the numbering stickers in agreement with case (a).
>
>After time 1:
> the final situation in case (a) is that the pot is empty.
> in case (b), the pot has infinitely many balls with no stickers!
>
>And yet at any intermediate time the two cases are indistinguishable!
>
>This sort of example shows that even omega-supertasks
>can be remarkably silly!


Actually, I think that they are fun to think about.
What this example (and similar ones) show is that
for tasks involving a transfinite number of steps,
you have to be more precise about what you are doing.
In order for mathematics to tell us what the result
is of performing some supertask, you have to describe
the supertask using standard mathematical objects
(typically sets). There are modeling choices that
have to be made, and the answer can depend on the
modeling choice. If it does depend on the modeling
choice, that means that the original supertask was
insufficiently specified.

The big ambiguity is how to compute limit states.
Associated with each ordinal alpha, there is a
corresponding state of the system, S_alpha. The
statement of the supertask explains how to go
from S_alpha to S_{alpha+1}. But that tells us
nothing about the state S_alpha when alpha is
a limit ordinal.

There are certain assumptions about the limit
states that are so "obvious" that they seem
to go without saying. For instance: "If a
ball is added at stage alpha_1, and is never
removed at any stage beta such that
alpha_1 < beta < alpha_2, and alpha_2 is a
limit ordinal, then the ball is present at
stage alpha_2. But that's an assumption about
the limit state. To really reason about supertasks,
you have to state all the assumptions about
limit states.




>
>> Now alter the situation slightly. At each step, again place 10 balls
>> into the vase and then remove one ball, but remove the ball
>> *randomly*. At the end, the vase may contain any number of balls

>
>Actually NOT. The pot will be empty(!) [with probability 1]
>
>For any ball, the probability of it being removed is like a harmonic
>series, which sums to oo, which by Borel-Cantelli means it will
>happen for sure. [meaning probability one, as always here]
>
>HOWEVER - if you add (say) 1, 4, 9, 16... balls per turn,
>and again remove one at random, each turn, then (Borel-Cantelli)
>each ball has a positive probability of being left behind.


Very interesting.

--
Daryl McCullough
Ithaca, NY



Date Subject Author
11/23/09
Read Re: Another AC anomaly?
Jesse F. Hughes
11/23/09
Read Re: Another AC anomaly?
Herman Jurjus
11/23/09
Read Re: Another AC anomaly?
master1729
11/25/09
Read Re: Another AC anomaly?
T.H. Ray
11/24/09
Read Re: Another AC anomaly?
george
12/1/09
Read Re: Another AC anomaly?
george
11/25/09
Read Re: Another AC anomaly?
Bill Taylor
11/26/09
Read Re: Another AC anomaly?
Daryl McCullough
11/30/09
Read What's wrong with supertasks? (was Re: Another AC anomaly?)
Herman Jurjus
12/1/09
Read Jesse Hughes of Bentley College uses signature block to defame
others; Ann Reeves Re: What's wrong with supertasks?
plutonium.archimedes@gmail.com
12/1/09
Read Re: Jesse Hughes of Bentley College uses signature block to defame
others; Ann Reeves Re: What's wrong with supertasks?
Marshall
12/1/09
Read using signature-blocks to defame innocent people Re: Jesse Hughes of
Bentley College
plutonium.archimedes@gmail.com
12/1/09
Read Re: Jesse Hughes of Bentley College uses signature block to defame
others; Ann Reeves Re: What's wrong with supertasks?
Seth Breidbart
12/1/09
Read Re: using signature-blocks to defame innocent people Re: Jesse Hughes
of Bentley College
Marshall
12/2/09
Read Re: using signature-blocks to defame innocent people Re: Jesse Hughes
of Bentley College
Marshall
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/27/09
Read Re: Another AC anomaly?
William Hughes
11/26/09
Read Re: Another AC anomaly?
mueckenh@rz.fh-augsburg.de
11/28/09
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mueckenh@rz.fh-augsburg.de
11/28/09
Read Re: Another AC anomaly?
Virgil
11/26/09
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William Hughes
11/28/09
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William Hughes
11/26/09
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mueckenh@rz.fh-augsburg.de
11/28/09
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ross.finlayson@gmail.com
11/28/09
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Virgil
11/29/09
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mueckenh@rz.fh-augsburg.de
11/29/09
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Virgil
11/29/09
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ross.finlayson@gmail.com
11/29/09
Read Re: Another AC anomaly?
Marshall
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
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ross.finlayson@gmail.com
11/30/09
Read Re: Another AC anomaly?
Virgil
11/30/09
Read Re: Another AC anomaly?
ross.finlayson@gmail.com
11/30/09
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Virgil
11/29/09
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mueckenh@rz.fh-augsburg.de
11/26/09
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LauLuna
11/26/09
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William Hughes
11/26/09
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anonymous.rubbertube@yahoo.com
11/29/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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Alan Smaill
11/29/09
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William Hughes
11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Alan Smaill
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Dik T. Winter
11/27/09
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mueckenh@rz.fh-augsburg.de
11/30/09
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Dik T. Winter
11/30/09
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mueckenh@rz.fh-augsburg.de
11/30/09
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Virgil
12/1/09
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Dik T. Winter
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
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Dik T. Winter
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
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William Hughes
12/1/09
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Virgil
12/1/09
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mueckenh@rz.fh-augsburg.de
12/1/09
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Virgil
12/1/09
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ross.finlayson@gmail.com
12/1/09
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Virgil
12/1/09
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ross.finlayson@gmail.com
12/1/09
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Virgil
12/2/09
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Dik T. Winter
12/2/09
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mueckenh@rz.fh-augsburg.de
12/2/09
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Dik T. Winter
12/2/09
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mueckenh@rz.fh-augsburg.de
12/2/09
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Virgil
12/3/09
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Dik T. Winter
12/3/09
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mueckenh@rz.fh-augsburg.de
12/3/09
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Dik T. Winter
12/3/09
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K_h
12/7/09
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Dik T. Winter
12/7/09
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Virgil
12/8/09
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K_h
12/8/09
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Virgil
12/9/09
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K_h
12/9/09
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Virgil
12/9/09
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mueckenh@rz.fh-augsburg.de
12/9/09
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Virgil
12/9/09
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K_h
12/10/09
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Dik T. Winter
12/9/09
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K_h
12/9/09
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Virgil
12/10/09
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Dik T. Winter
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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Virgil
12/10/09
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Dik T. Winter
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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Virgil
12/15/09
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Dik T. Winter
12/15/09
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mueckenh@rz.fh-augsburg.de
12/15/09
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Dik T. Winter
12/15/09
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K_h
12/16/09
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Virgil
12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
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Virgil
12/17/09
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Dik T. Winter
12/17/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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Virgil
12/17/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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YBM
12/17/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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Quaestor
12/18/09
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mueckenh@rz.fh-augsburg.de
12/18/09
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Virgil
12/18/09
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Dik T. Winter
12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
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Virgil
12/22/09
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12/22/09
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Virgil
12/18/09
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Dik T. Winter
12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
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Quaestor
12/21/09
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Dik T. Winter
12/21/09
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mueckenh@rz.fh-augsburg.de
12/21/09
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Marshall
12/21/09
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Virgil
12/22/09
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Dik T. Winter
12/27/09
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mueckenh@rz.fh-augsburg.de
12/27/09
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Virgil
12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
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Virgil
12/11/09
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K_h
12/11/09
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Dik T. Winter
12/11/09
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K_h
12/11/09
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Marshall
12/12/09
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Jesse F. Hughes
12/12/09
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K_h
12/12/09
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K_h
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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Virgil
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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Virgil
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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Virgil
12/10/09
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Dik T. Winter
12/10/09
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Dik T. Winter
12/11/09
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K_h
12/11/09
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Virgil
12/7/09
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mueckenh@rz.fh-augsburg.de
12/7/09
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Virgil
12/8/09
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Dik T. Winter
12/8/09
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mueckenh@rz.fh-augsburg.de
12/8/09
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anonymous.rubbertube@yahoo.com
12/8/09
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Virgil
12/8/09
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mueckenh@rz.fh-augsburg.de
12/8/09
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Virgil
12/10/09
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Dik T. Winter
12/3/09
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Virgil
12/3/09
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mueckenh@rz.fh-augsburg.de
12/3/09
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Virgil
12/7/09
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Dik T. Winter
12/7/09
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Virgil
12/7/09
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mueckenh@rz.fh-augsburg.de
12/7/09
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Virgil
12/8/09
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Dik T. Winter
12/8/09
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mueckenh@rz.fh-augsburg.de
12/8/09
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Virgil
12/10/09
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Dik T. Winter
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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Virgil
12/10/09
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Dik T. Winter
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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Virgil
12/11/09
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Virgil
12/15/09
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Dik T. Winter
12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
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Dik T. Winter
12/16/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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T.H. Ray
12/17/09
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Dik T. Winter
12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
12/4/09
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Marshall
12/7/09
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Dik T. Winter
12/7/09
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mueckenh@rz.fh-augsburg.de
12/7/09
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Virgil
12/8/09
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Dik T. Winter
12/8/09
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mueckenh@rz.fh-augsburg.de
12/8/09
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Virgil
12/10/09
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Dik T. Winter
12/10/09
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mueckenh@rz.fh-augsburg.de
12/10/09
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Virgil
12/10/09
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Dik T. Winter
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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Marshall
12/11/09
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mueckenh@rz.fh-augsburg.de
12/11/09
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Virgil
12/11/09
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Virgil
12/11/09
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Marshall
12/12/09
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mueckenh@rz.fh-augsburg.de
12/12/09
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Virgil
12/12/09
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Marshall
12/12/09
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mueckenh@rz.fh-augsburg.de
12/12/09
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Virgil
12/12/09
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Marshall
12/12/09
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mueckenh@rz.fh-augsburg.de
12/12/09
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Virgil
12/12/09
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Marshall
12/12/09
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george
12/12/09
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Virgil
12/12/09
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george
12/13/09
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mueckenh@rz.fh-augsburg.de
12/13/09
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Virgil
12/15/09
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Dik T. Winter
12/16/09
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mueckenh@rz.fh-augsburg.de
12/16/09
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Virgil
12/17/09
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Dik T. Winter
12/17/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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Quaestor
12/17/09
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mueckenh@rz.fh-augsburg.de
12/17/09
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Quaestor
12/18/09
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mueckenh@rz.fh-augsburg.de
12/18/09
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Virgil
12/18/09
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Dik T. Winter
12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
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Virgil
12/19/09
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mueckenh@rz.fh-augsburg.de
12/19/09
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Marshall
12/19/09
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Virgil
12/19/09
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Virgil
12/19/09
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ross.finlayson@gmail.com
12/19/09
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Virgil
12/22/09
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ross.finlayson@gmail.com
12/22/09
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Marshall
12/27/09
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ross.finlayson@gmail.com
12/21/09
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Dik T. Winter
12/21/09
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mueckenh@rz.fh-augsburg.de
12/21/09
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Virgil
12/22/09
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Dik T. Winter
12/27/09
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mueckenh@rz.fh-augsburg.de
12/27/09
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Marshall
12/27/09
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Virgil
12/27/09
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Virgil
12/27/09
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Ralf Bader
12/27/09
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Virgil
12/13/09
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mueckenh@rz.fh-augsburg.de
12/13/09
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Virgil
12/4/09
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K_h
12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
12/4/09
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mueckenh@rz.fh-augsburg.de
12/4/09
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Virgil
12/5/09
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mueckenh@rz.fh-augsburg.de
12/5/09
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12/5/09
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mueckenh@rz.fh-augsburg.de
12/6/09
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Virgil
12/5/09
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Carsten Schultz
12/2/09
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Virgil
12/1/09
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george
12/1/09
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Virgil
11/27/09
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Virgil
11/27/09
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anonymous.rubbertube@yahoo.com
11/30/09
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William Hughes
11/27/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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William Hughes
11/27/09
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anonymous.rubbertube@yahoo.com
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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William Hughes
11/27/09
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mueckenh@rz.fh-augsburg.de
11/27/09
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Virgil

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