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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
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
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.
Jul
5
comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”
@ajotatxe: did you learn integration before you learned this result?
Jul
5
comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$”
@MichaelHardy: I suppose that one can use either the area or the circumference of the unit circle as the basis for the definition of $\pi$. But then one would need to show that the same number of this definition reappears in the post's theorem's formula.
May
25
comment Looking for a “Guide for the Perplexed by Low-dimensional Topology”
@GregoryGrant: At least part of the reason I find this stuff unsettling is that pretty much everywhere else (outside of areas devoted to these dimension-specific peculiarities) the sets $\mathbb{R}^n$ are treated as "basically the same" modulo a "trivial" difference in the dimensions. In particular, people routinely reason about "$\mathbb{R}^n$" (for some unspecified $n$) by analogy with their intuitions about $\mathbb{R}^d$ for $d \in \{1, 2, 3\}$. If every dimension has its own quirks, such thinking by analogy (where the target of the analogy is wholly unfamiliar) becomes highly suspect.
May
16
comment Conditions for “equal cardinality $\Rightarrow$ equal (finite) measure”?
@PhoemueX: I've edited my post in response to your comment.
May
12
comment On random rotational fluctuations in $\mathbb{R}^n$
Thanks for posting this! It is indeed beyond my grasp, as you predicted. Since it's probably bad form to accept an answer that one can't understand, I won't until I do, but I'm working on it! I will use your answer as something to aim for. Thanks again.
May
12
comment On random rotational fluctuations in $\mathbb{R}^n$
@StephenMontgomery-Smith: thanks, please post your answer; it will give me something concrete to aim for as I read on stochastic diff eqs (I started already).
May
11
comment On random rotational fluctuations in $\mathbb{R}^n$
@StephenMontgomery-Smith: unfortunately no. I'm willing to learn it, but it will probably take me a while.
Sep
12
comment Gaussian proof for the sum of squares?
Spectacular! Thanks for posting it.
Aug
17
comment What's the name of $\sum_{k = 0}^{n} (-1)^k {n \choose k} (n-k)^w$?
@PedroTamaroff: thanks! Your comment pointed me in the right direction. According the the Wikipedia page on the I-E principle, the expression I found equals $n! \, S(w, n)$, where $S(w, n)$ is a "Stirling number of the second kind"...
Mar
17
comment Summation involving totient function: $\sum_{d\mid n} \varphi(d)=n$
@Sabyasachi: sorry, I had not finished with my comment. Also, on the contrary, I think your proof is extremely elegant. I'm making the case for inelegance here.
Mar
17
comment Summation involving totient function: $\sum_{d\mid n} \varphi(d)=n$
$\dots\;\;$ These means that $\nset$ can be partitioned according to the $d$ factor of these products, and the size of each partition is $\varphi(n/d)$. But don't get me wrong: such proofs are OK, as quick ways to convince oneself that something is true, but one needs to go beyond them to get at the essence of why something is true.
Mar
17
comment Summation involving totient function: $\sum_{d\mid n} \varphi(d)=n$
What sticks in my craw about this argument is that fractions are really inessential to the problem; they get brought in only because they come with a built-in, widely recognized "simplified to lowest terms" operation. IOW, fractions show up in this theorem only as, basically, a cute hack, and IMO they obscure what's really happening in the theorem, namely that (1) every $m \in \def\nset{\{1,\dots,n\}}\nset$ can be written uniquely as the product $du$ where $d\mid n$ and $\gcd(u, n/d) = 1$, and (2) every such product $du$ belongs to $\nset$. $\;\;\dots$
Mar
16
comment On last digit of 4 consecutive primes less than 10 apart
I see. Thanks for the clarifiication!
Mar
16
comment Artin 2nd Ed. Problem 12.5.3
As a follow-up to Ted Shifrin's comment, you may find this answer useful.