Recent content by CaffeineJunky

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    Prove 29 is not irreducible in Z

    Why wouldn't it be it? Goal: Show that 29 can be factored over the Gaussian integers. Proof (by explicit example): (5 - 2i)(5 + 2i) = 25 - 10i + 10i - 2i^2 = 25 + 4 = 29. QED
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    Undergrad Solve the Pythagorean Tripple 3^x+4^x=5^x

    Fermat-Wiles theorem. Give the guy credit who proved it along with the one who conjectured it.
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    High School How do you use the quotient rule to find the derivative of X?

    Use the product rule or the quotient rule.
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    Are the Rows of an Invertible Matrix Linearly Independent?

    Yes. Keep in mind that \det(A) = \det(A^{T}), hence every portion of the invertible matrix theorem automatically applies to the rows as well as the columns. You should come to see that there is a relationship between the row space and the column space of a matrix, along with the null space...
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    Graduate Periodic, continuous and piecewise smooth function

    sin(x) is a continuous smooth piecewise function.
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    Graduate What Is a Lie Group Without an Identity Matrix Called?

    If there is no identity then there are no inverses, so at best you'll be left with a semigroup.
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    Graduate Why is (0,1) not compact in topology?

    The proof is the Heine-Borel theorem.
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    Prove that if x > limsup s_n, then x is not the limit of any subsequence

    I'm not sure if the proof is 100% correct. I mean, I did word it poorly and things do need to be fixed up. but I believe the general idea is correct. 1. It just seemed to fit in order to find a value of epsilon: Take the smallest element of the set and take a smaller element to be your epsilon...
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    Prove that if x > limsup s_n, then x is not the limit of any subsequence

    This should fall into place by using both facts. Try using a contradiction. I have a rough proof below: WARNING: A POSSIBLE SOLUTION IS HERE. DO NOT LOOK UNLESS YOU WANT TO. Suppose x > lim sup (S_n) and some subsequence (S_K(n)) has x as a limit, where K(n) is a strictly increasing function...
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    Uniqueness of Even and Odd Function Decomposition

    Yes, you're logic is essentially the same as using the suggestion about F(x) + F(-x) and F(x) - F(-x).
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    Uniqueness of Even and Odd Function Decomposition

    Your proof is nearly there, although the wording can be cleaned up. There is a real simple solution that you are passing up that allows you to avoid using a pointwise argument and that involves simply using F(x) and F(-x), and noting a certain property about F(x) + F(-x) and F(x) - F(-x).
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    Uniqueness of Even and Odd Function Decomposition

    How do you know you can always define functions O and E such that O is odd, E is even, and F = O + E? i.e. Is it possible for a function F \neq O + E for all choices of odd functions O and even functions E? (The answer is no, but you assume that O and E will always exist without proof. This...
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    Uniqueness of Even and Odd Function Decomposition

    Your proof of A is still bad because you're still assuming that two functions E and O exist and their sum is F, where F is our arbitrary function. This defeats the purpose of showing that the functions E and O exist, since you already assume their existence. (It's easy to prove A is true when...
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    Uniqueness of Even and Odd Function Decomposition

    Ok, I see what you did there. But you can avoid the contradiction and simply prove the theorem straightforward by using the case by case method, except for the case where F is neither even nor odd (you'd be exploiting the fact that the set of functions F that satisfy the property being...
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    Uniqueness of Even and Odd Function Decomposition

    This is wrong. If F is an even function, then 0 = 2O is perfectly reasonable, seeing as O is necessarily the zero function. (If O were not the zero function, then F wouldn't be even, since F = E + O would be neither odd nor even.) (But then again, I don't quite get what you were trying to prove...