kjo
Reputation
4,811
Next privilege 5,000 Rep.
Approve tag wiki edits
 Jul 26 accepted On the importance of natural transformations Jul 23 awarded Yearling Jul 14 comment On the importance of natural transformations Also, as I think more about your answer, I'd say that, in a way, it just reiterates what I wrote in my question: the definition of natural transformation adds nothing to what I already knew. (It's like learning that one has been "speaking prose all one's life without knowing it". What does one do with this fact?) I don't think I ever saw the proof of the fact that the complex eigenvalues of a matrix in $\mathrm{GL}_n\mathbb{R}$ are also eigenvalues of the same matrix when viewed as belonging to $\mathrm{GL}_n\mathbb{C}$, but this fact strikes me as more or less obvious. Jul 14 comment On the importance of natural transformations Thanks. I think I get the gist of your answer, but I'm confused by your use of the expression $\mathbb{R}[t] \hookrightarrow \mathbb{C}[t]$, and more specifically, the trailing $[t]$'s. If I were to write out how I understand your answer, I would have written something beginning with: "let $f$ be the insertion of $\mathbb{R} \hookrightarrow \mathbb{C}$", etc. Am I right? Jul 14 revised On the importance of natural transformations added 2 characters in body Jul 14 revised On the importance of natural transformations added 10 characters in body Jul 14 revised On the importance of natural transformations added 113 characters in body Jul 14 revised On the importance of natural transformations added 113 characters in body Jul 14 revised On the importance of natural transformations deleted 72 characters in body Jul 14 asked On the importance of natural transformations Jul 11 awarded Disciplined Jul 6 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” The suitability of this approach, then, hinges on the details of the proof of $A = \frac{1}{2} rC$... Since you don't give them, I assume that they are very straightforward, but I can't find a particularly simple proof of this assertion. In any case, thanks for posting this strategy. Jul 6 revised What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane? added 19 characters in body Jul 6 accepted What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane? Jul 5 asked What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane? Jul 5 revised Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” fixed typo in (3) Jul 5 accepted Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” Jul 5 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” Thanks, that's an elegant rendition. Jul 5 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” Thank you. Even though, as you point out, this proof is not complete, and therefore (rigorously speaking) it isn't rigorous, it contains the essentials of what I was after. It does a very good job of demarcating those bits that may really require non-elementary methods from those that don't. Jul 5 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” @DavidC.Ullrich: I've been intentionally vague on the points you mention precisely because I don't know what's possible. For example, AFAIK, at least as of not too long, the only proof known of Fermat's Last Theorem relied on mathematics established well after Fermat's time. For all I know, something similar is true of the theorem in this post. In that case, I'm interested in a proof that requires the least additional mathematics than was know by the ancients who were already familiar with the theorem.