Forum: Puzzles
2017-06-03, 20:07
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Replies: 10
Views: 1,050
1) Well \mathbb{Q} (the set of rationals) has...
1) Well \mathbb{Q} (the set of rationals) has height \aleph_1, despite being countable. One can show by transfinite induction that every countable \alpha embeds into \mathbb{Q}, and taking the \sup...
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Forum: Puzzles
2017-06-02, 19:41
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Replies: 10
Views: 1,050
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Forum: Puzzles
2017-06-01, 19:47
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Replies: 10
Views: 1,050
Heights of Models of PA
This question is inspired by an incomplete solution of mine to a final exam problem a few years back.
For M a countable model of PA, let \text{height}(M)=\sup\{\alpha:\exists f:\alpha\rightarrow...
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Forum: News
2016-02-01, 09:41
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Replies: 631
Views: 68,898
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Forum: News
2016-01-23, 18:43
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Replies: 631
Views: 68,898
Well I can't find it at all by browsing the...
Well I can't find it at all by browsing the relevant subforum. I mean, the last post is 29 Mar 2013, but I can't find a thread whose last post is that just by browsing.
Was only able to find it...
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Forum: News
2016-01-23, 18:21
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Replies: 631
Views: 68,898
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Forum: Homework Help
2015-03-20, 17:51
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Replies: 2
Views: 1,537
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Forum: Abstract Algebra & Algebraic Number Theory
2013-11-25, 09:01
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Replies: 2
Views: 3,417
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Forum: Abstract Algebra & Algebraic Number Theory
2013-11-25, 04:48
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Replies: 2
Views: 3,417
Discrete Ordered Rings?
Are there any examples other than \mathbb{Z}?
(Alright, alright. I know of one other example: \prod_U\mathbb{Z}, i.e. ultrapowers of [tex]\mathbb{Z}[tex]. But this is not helpful for the problem...
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Forum: Other Mathematical Topics
2013-05-25, 08:35
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Replies: 11
Views: 2,457
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Forum: Other Mathematical Topics
2013-05-23, 07:11
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Replies: 11
Views: 2,457
How about a concrete example? Take M to be the...
How about a concrete example? Take M to be the closed unit disc in the complex plane. Then its boundary is the unit circle. Taking U = {re^it: r>1/2}, we get a diffeomorphism of U with S^1 x [0,1)...
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Forum: Other Mathematical Topics
2013-05-21, 22:25
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Replies: 11
Views: 2,457
Here's the partition of unity theorem:
Let X...
Here's the partition of unity theorem:
Let X be a manifold. For any covering of X by open sets {U_alpha}, there is a sequence of smooth functions {theta_i} on X such that:
(a) 0 <= theta_i <= 1...
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Forum: Other Mathematical Topics
2013-05-21, 13:15
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Replies: 11
Views: 2,457
Manifolds Question
Let M be a compact n-manifold with boundary. Show that there is an open set U containing the boundary that is diffeomorphic to (the boundary) x [0,1)
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By definition of the boundary of a...
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Forum: Puzzles
2013-03-20, 12:57
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Replies: 4
Views: 1,342
Thanks for the links. I already read Journey...
Thanks for the links. I already read Journey through Genius in high school; that's where I learned of this solution!
Going to have to think about the Weierstrass Factorization and it's application...
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Forum: Puzzles
2013-03-20, 01:05
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Replies: 4
Views: 1,342
Can Euler's Basel Solution be Saved?
I'm thinking of giving a talk to an undergrad math club, and I had the idea of presenting Euler's solution to the Bessel Problem (1 + 1/4 + 1/9 + ... = pi^2/6) as the topic. While there are some very...
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Forum: Lounge
2013-02-20, 07:14
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Replies: 5
Views: 612
Higgs Boson and End of Universe?
This sounds interesting...
http://www.livescience.com/27218-higgs-boson-universe-future.html
But, I don't really trust popular science article to get something like this right. For some reason,...
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Forum: Lounge
2013-02-15, 22:06
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Replies: 205
Views: 9,266
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Forum: Lounge
2013-02-15, 21:43
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Replies: 205
Views: 9,266
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Forum: Lounge
2013-02-15, 05:04
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Replies: 205
Views: 9,266
Wish granted. In order to ensure that ethanol...
Wish granted. In order to ensure that ethanol preparations have no effect at all after the 2-3 hour buzz, including social effects and property damage caused by acting foolishly, you lock yourself...
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Forum: Abstract Algebra & Algebraic Number Theory
2013-01-02, 18:09
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Replies: 5
Views: 4,682
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Forum: Abstract Algebra & Algebraic Number Theory
2013-01-02, 06:17
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Replies: 5
Views: 4,682
Thanks. I presume 'rational integer' and...
Thanks. I presume 'rational integer' and 'rational prime' mean 'element of \mathbb{Q}' and 'prime in \mathbb{Q}' respectively? As opposed to 'element of \mathbb{O_K}' and 'prime in \mathbb{O_K}'?
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Forum: Abstract Algebra & Algebraic Number Theory
2012-12-31, 06:22
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Replies: 5
Views: 4,682
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Forum: Lounge
2012-07-12, 09:47
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Replies: 14
Views: 870
Just checking in to this thread again...
...
Just checking in to this thread again...
Perhaps I should have been even more explicit. I'm looking for the name of a specific test that allows one to accept or reject the null hypothesis, which...
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Forum: Lounge
2012-07-06, 11:00
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Replies: 14
Views: 870
Well, that was fun, and a bit tiring. I didn't...
Well, that was fun, and a bit tiring. I didn't know about the 10,000 character per post limit, which prevented me from asking my question in the original post. Anyway, my question was this: Is there...
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Forum: Lounge
2012-07-06, 10:47
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Replies: 14
Views: 870
Statics Question, in the form of a storey
--- Part 1 ---
Alice: Hello Bob. So glad you could spare a few minutes out of your busy day.
Bob: No problem Alice. It's on my way back home anyway. Now what did you ask me to come for again?...
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