Billy
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 Jul 28 comment what is the difference between sections and germs in a sheaf? Sections are "functions on a large open set". Germs are "functions around a point". If this isn't obvious, why don't you tell us what your definitions are? Jul 28 comment Fixed point of a continuous map For the first problem, consider the map $g:x\mapsto f(x) - x$ on $[-1, 1]$. What is $g(0)$? What is $g(1)$? Where does $f$ have a fixed point? Jul 28 comment Help to prove bijection between subset of $S^2$ and $\mathbb{R}^2$ @user1620696 No problem! Jul 28 revised Help to prove bijection between subset of $S^2$ and $\mathbb{R}^2$ added 655 characters in body Jul 28 comment Help to prove bijection between subset of $S^2$ and $\mathbb{R}^2$ I've edited my post and added the surjectivity argument. Jul 28 answered Help to prove bijection between subset of $S^2$ and $\mathbb{R}^2$ Jul 28 comment Find equation that represents $m$ as the subject @Nick The answer is, essentially, that you need to plug numbers into formulas, or perhaps write a computer program to do so if you can't face doing it yourself. If you try to solve it algebraically, you'll end up with something disgusting like George's formulas above no matter how you spin it (Wikipedia's methods are all different ways of saying the same thing). I like George's recursive algorithm, but of course it (almost certainly) won't give you exact answers. Still, let me know if I can help with anything. Jul 27 comment Find equation that represents $m$ as the subject Perhaps the most appropriate tag is "algebra-precalculus". (Though I wouldn't expect anyone to learn how to solve a cubic equation until well after calculus, admittedly, but it has the same flavour as a quadratic equation, just much uglier...) Jul 27 answered Find equation that represents $m$ as the subject Jul 27 comment Why do I disagree with my calculator? I agree with GEdgar, but if remembering that numbers have to be packaged with the sign to the left of them is hard, it's enough to remember that + and - are to be given the same priority, and so you should work from left to right, which always gives the right answer. Jul 27 comment How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole Ah, sorry - my three-dimensional mental picture confused me. Thanks for the clarification. Jul 27 comment How do I rotate a given point on $\mathbb{S}^n$ so that I send it to the South Pole But all rotations that send P to S do arise in this way, right? Jul 27 comment Why do real powers need positive bases? (and to handle them) You are absolutely right. It is not true that the power $a^b$ is only well-defined when $a\geq 0$. You can ascribe a well-defined value to $a^b$ for any $a$ when $b$ is a positive integer, for example; in fact, you can extend the function far more than that. But you lose a lot of its nice properties (e.g. $a^{b_1} a^{b_2} = a^{b_1+b_2}$) when you do so. It's fairly safe to say that the power $a^b$ is only defined for all $b$ and in a nice way when $a\geq 0$. Jul 27 awarded Quorum Jul 27 comment how to interpret theorem about polynomial factorization over modulo ring? Have you misunderstood the result? You are given a formula - $a_n(1 + \sum a_i)$ - and told that, if it's $0$, then your polynomial has a factor, and if it's not $0$, then it doesn't. There is no need to distinguish between trivial and non-trivial linear factors, because the result is true in both cases. $X+1$ is a factor of $X+1$, even if you want to think of it as somehow "trivial". Jul 27 comment how to interpret theorem about polynomial factorization over modulo ring? (What is a "trivial" linear factor anyway? There are only two linear polynomials - $X$ and $X+1$ - and neither satisfies the condition $a_n(1 + \sum a_i) \neq 0$.) Jul 27 comment how to interpret theorem about polynomial factorization over modulo ring? The claim is that that polynomial has no linear factors if and only if $a_n(1 + \sum a_i) \neq 0$. In the case of $X+1$, $a_n(1 + \sum a_i) = 1(1+1) = 0$. So you should expect a linear factor. This linear factor is $X+1$. Where's the confusion? Jul 27 answered What character can replace word “let” in proofs? Jul 27 comment Arithmetic Base Conversion If x is an even number (except 2), write 1 + (x-1). If x is an odd number (except 1, 3, 5, 7), write 1 + 3 + (x-4). Or, if you like, x is already a sum of one odd number (x). Jul 23 comment Prove $\sqrt{k}$ is not a rational number. Sorry, but this is nonsense. The English doesn't even make sense. Can you rewrite your proof carefully?