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Dec
1
comment Why is the area of the circle $πr^2$?
Actually, Archimedes used a method of exhaustion: en.wikipedia.org/wiki/Area_of_a_disk#Archimedes.27s_proof
May
21
comment showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
Let $\phi$ denote the monomorphism. Then exactness at $FA$ tells us $\ker Fg = \operatorname{im} \phi$. And then maybe it helps if we identify $\operatorname{im} \phi$ with $F(\ker f)$? (Maybe part of my confusion is due to the fact that I'm still used to thinking of kernels and images as objects?)
May
21
comment showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)
Hmm, exactness at $F(\ker f)$ means that $F(\ker f) \to FA$ is a monomorphism. Our goal is to show that $\ker Fg = \operatorname{im} Ff$, so we need to extract some information about $\ker Fg$ from the monomorphism $F(\ker f) \to FA$, somehow?
Jun
26
comment How to show that this function (related to the zeta function) is even?
Sorry, what do you mean by "harmonic sum"? A sum that arises when dealing with Fourier series?
Jun
25
comment How to show that this function (related to the zeta function) is even?
Wow, thanks! Could you explain how you were able to come up with all of that?
Apr
11
comment Transpose of a linear operator on functions
Ah, thanks! I was thinking of using the divergence theorem, but I wasn't sure if it was applicable (partly because I didn't know what was the exact space of functions $L$ was acting on).
Jan
22
comment Why is $y = \sqrt{x-4}$ a function? and $y = \sqrt{4 - x^2}$ should be a circle
@vincentbelkin, I edited my answer.
Jan
22
comment Why is $y = \sqrt{x-4}$ a function? and $y = \sqrt{4 - x^2}$ should be a circle
@AsafKaragila, thanks!
Sep
1
comment Is there a simple way to bound this contour integral?
Thanks for your comment. I realized that I was having a hard time phrasing my question. Now that I think about it, I wanted to be able to get a sharper estimation - I wanted to be able to get a feel for how $\int e^{-R^2 \cos2\theta} R \, d\theta$ behaves for large values of $R$ (even though to evaluate the original integral, we just needed to show this was $o(1)$). But like you said, this estimation isn't too bad. How did you get that estimate of yours?
Sep
1
comment Bounding a function by its second derivative using Fourier series
I am sorry, but I am confused. I am asking if there is a way to bound a function by its second derivative with Fourier series, not when a function has a Fourier series.
Aug
9
comment Proving the mean value property of harmonic functions using distributions?
Thank you! This was really easy to follow! One question I have is: how do we generalize this argument to higher dimensions? I know the radial component of $\Delta \phi$ will be $\phi_{rr}+\frac{n-1}{r}\phi_r$, but why is the integral of the remaining part zero?
Jan
20
comment Fundamental domain for the group of transformations generated by $\tau \mapsto \tau + 2$ and $\tau \mapsto -1/\tau$
Thank you! I had not thought about the structure of $G$ at all.
Sep
8
comment How to show that $\lim\limits_{x \to \infty} f'(x) = 0$ implies $\lim\limits_{x \to \infty} \frac{f(x)}{x} = 0$?
This is really nice! Thanks!
Sep
8
comment How to show that $\lim\limits_{x \to \infty} f'(x) = 0$ implies $\lim\limits_{x \to \infty} \frac{f(x)}{x} = 0$?
Wow! This really is an immediate consequence of L'Hopital's rule! Thanks for helping me finish my solution!
Aug
9
comment How do I prove the divergence of this series?
@Didier: Oh yeah, thanks for pointing that out!
Aug
9
comment How do I prove the divergence of this series?
@Asaf: Hmm, that's a good point. If the original problem was with $\frac{1}{\log log n}$, I guess my solution was overkill.
Jun
8
comment How does the parallelogram law imply the existence of an inner product for a given norm?
Thanks! I'm sorry I didn't find that when I searched for this question.
May
27
comment Understanding the Schwarz reflection principle
Thanks! That was the statement and construction given by the book, and while I could understand it, I didn't understand why my construction didn't work.
May
27
comment Understanding the Schwarz reflection principle
Thank you! This was exactly my problem! I had assumed that $f(\overline{z})$ was holomorphic because it was a pretty "nice and simple" function. I just started studying complex analysis, so I had the definition of holomorphic memorized without much intution, but I understand the idea a lot better now!
Jan
16
comment Defining the determinant of linear transformations as multilinear alternating form
Aha thanks! I wonder how I overlooked the linearity of $T$. Hm... but I'm also wondering if this could be done in a way similar to the Steinitz exchange lemma. Suppose we order the $e_i$'s and $f_i$'s so that for all $0 \leq k \leq n$, $f_1, \ldots, f_k, e_{k+1}, \ldots, e_n$ span $V$. Then we could go from $vol(Te_1, \ldots, Te_n)/vol(e_1, \ldots, e_n)$ to $vol(Tf_1, \ldots, Tf_n)/vol(f_1, \ldots, f_n)$, replacing one vector at a time. (And the reason the ratio stays the same is because $T$ is linearity.)