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 Post subject: plus or minus one questionPosted: Tue, 22 May 2012 16:16:07 UTC
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Wasn't sure if this was the right forum for this problem.~

We are given a -tuple - , where is equal to either or to . We transform this -tuple by replacing the original -tuple with .
Prove that after a finite number of transformations we get .

Hmmm I found an induction solution but I was wondering if there was a more tricky solution (possibly monovariants)?

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 Post subject: Re: plus or minus one questionPosted: Tue, 22 May 2012 16:54:46 UTC
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rdj5933mile5math64 wrote:
Wasn't sure if this was the right forum for this problem.

We are given a -tuple - , where is equal to either or to . We transform this -tuple by replacing the original -tuple with .
Prove that after a finite number of transformations we get .

Hmmm I found an induction solution but I was wondering if there was a more tricky solution (possibly monovariants)?

My solution: The desired result is equivalent to asserting the circulant matrix over , where is the so-called fundamental circulant matrix of order , is nilpotent. Now using the Frobenius homomorphism (and induct on ), we have , as claimed.

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 Post subject: Re: plus or minus one questionPosted: Tue, 22 May 2012 18:12:51 UTC
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outermeasure wrote:
My solution: The desired result is equivalent to asserting the circulant matrix over , where is the so-called fundamental circulant matrix of order , is nilpotent. Now using the Frobenius homomorphism (and induct on ), we have , as claimed.

O.O whaaaaa? I'm not sure what most of this means.~ I looked up circulant matrices on wikipedia, but I have no clue how one can relate your first statement, to the original question. Hmmmmm I should learn algebra.

On a different note, I fail - I found out that there is a non-inductive solution (in addition to outermeasure's solution). Well, it depends on whether or not you count a certain combinatorial identity as proven by induction or not.

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 Post subject: Re: plus or minus one questionPosted: Tue, 22 May 2012 23:53:31 UTC
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rdj5933mile5math64 wrote:
outermeasure wrote:
My solution: The desired result is equivalent to asserting the circulant matrix over , where is the so-called fundamental circulant matrix of order , is nilpotent. Now using the Frobenius homomorphism (and induct on ), we have , as claimed.

O.O whaaaaa? I'm not sure what most of this means.~ I looked up circulant matrices on wikipedia, but I have no clue how one can relate your first statement, to the original question. Hmmmmm I should learn algebra.

On a different note, I fail - I found out that there is a non-inductive solution (in addition to outermeasure's solution). Well, it depends on whether or not you count a certain combinatorial identity as proven by induction or not.

Basic abstract algebra is commonly found in college-level mathematics, it's not unusual that such a solution would not occur to you.

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 Post subject: Re: plus or minus one questionPosted: Wed, 23 May 2012 02:58:33 UTC
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Joined: Wed, 4 Apr 2012 03:51:40 UTC
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Basic abstract algebra is commonly found in college-level mathematics

But all of my friends are doing it.

The only -tuples that become are the ones described above. It's proven by working backwards.

Now, the idea behind my *improved* solution is to look at
Spoiler:

and

Spoiler:

The original solution only used induction and the first part.

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 Post subject: Re: plus or minus one questionPosted: Wed, 23 May 2012 03:18:47 UTC
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rdj5933mile5math64 wrote:
But all of my friends are doing it.

It sounds as if you need a much more diverse group of friends.

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 Post subject: Re: plus or minus one questionPosted: Wed, 23 May 2012 06:45:48 UTC
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rdj5933mile5math64 wrote:
outermeasure wrote:
My solution: The desired result is equivalent to asserting the circulant matrix over , where is the so-called fundamental circulant matrix of order , is nilpotent. Now using the Frobenius homomorphism (and induct on ), we have , as claimed.

O.O whaaaaa? I'm not sure what most of this means.~ I looked up circulant matrices on wikipedia, but I have no clue how one can relate your first statement, to the original question. Hmmmmm I should learn algebra.

On a different note, I fail - I found out that there is a non-inductive solution (in addition to outermeasure's solution). Well, it depends on whether or not you count a certain combinatorial identity as proven by induction or not.

Write your tuples of as and look at --- a vector over due to the ambiguity in the choice of from . The effect of your mapping on is . Hence the statement every tuple will eventually end up as is equivalent to saying for all and large enough --- which is the same as saying is nilpotent in .

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 Post subject: Re: plus or minus one questionPosted: Thu, 24 May 2012 17:50:32 UTC
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Joined: Wed, 4 Apr 2012 03:51:40 UTC
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Oh ok I understand now! Thanks outermeasure!

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