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May
9
comment Show that $\sum_{n=0}^{\infty}\frac{1}{4n^4+1}=\frac{1}{2}+\frac{\pi}{4}\tanh\left(\frac{\pi}{2}\right)$
Duplicate of Closed form for $\sum_{n=-\infty}^{\infty} \frac{1}{n^4+a^4}$
May
9
comment Showing that $\pi(M \# N) = \pi(M) \ast \pi(N)$ for $n$-dimensional manifolds $M$,$N$
$\pi_1(M)$ is a group not a number, so «$\pi_1(M)=\infty$» is not even wrong
May
9
comment Showing that $\pi(M \# N) = \pi(M) \ast \pi(N)$ for $n$-dimensional manifolds $M$,$N$
Hint: use Seifert–van Kampen
May
9
comment Integrals in analysis and category theory
(Somewhat) related: «...will instantly recognise this formula as a tropical integral...» @ MO
May
8
comment Prove that $\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k = 1}^{n}{f(\frac{k}{n}) }$ $=\int_0^1 f(x)dx.$
Well, yes, if Riemann integral exists (in your example not Riemann integral, but only improper integral exists).
May
8
comment Prove that $\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k = 1}^{n}{f(\frac{k}{n}) }$ $=\int_0^1 f(x)dx.$
@Brian AFAICS Martin answered the question completely a couple of months ago, no?
May
8
comment Prove that $\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k = 1}^{n}{f(\frac{k}{n}) }$ $=\int_0^1 f(x)dx.$
Dear robjohn, (while the answer is technically correct) the result has nothing to do with uniform continuity — the statement is true even if $f$ is not cont. (as long as the RHS exists).
May
8
comment Prove that $\lim_{n\rightarrow\infty} \frac{1}{n}\sum_{k = 1}^{n}{f(\frac{k}{n}) }$ $=\int_0^1 f(x)dx.$
Well, what is the definition of integral [you're using]?
May
8
comment A question on Jacobi sums
Related: Explicit formula for Fermat's 4k+1 theorem
May
8
comment At least $p^2-p$ solutions to $x^2+y^2+z^2 \equiv 1 \mod p$
@WillJagy Yes, and it's not hard to prove this answer using Jacobi sums.
May
8
comment Complex structure on $\mathbb{C}\mathbb{P}^2\# \dots \# \mathbb{C}\mathbb{P}^2$
@mezhang Connected sum (the notation is absolutely standard)
May
8
comment Complex structure on $\mathbb{C}\mathbb{P}^2\# \dots \# \mathbb{C}\mathbb{P}^2$
related: How can we detect the existence of almost-complex structures? @ MO
May
8
comment Complex structure on $\mathbb{C}\mathbb{P}^2\# \dots \# \mathbb{C}\mathbb{P}^2$
related: Almost complex structure on $\mathbb CP^2 \mathbin\# \mathbb CP^2$
May
7
comment Showing that $\pi(G/H, 1) = H$ under a condition
$G$ is the universal cover of $G/H$.
May
7
comment $p$ is an odd prime number where $p=3k+1\Longleftrightarrow\exists a,b\in\Bbb Z^+$ such that $p=a^2+ab+b^2$
possible duplicate of Primes congruent to 1 mod 6
May
7
comment mapping in $H_{c}^k(X,\mathbb{Z})$
related: Long exact sequence for cohomology with compact supports
May
6
comment Has this variation of Hochschild cohomology been studied?
This reminds me of an interpretation of Gerstenhaber-Schack cohomology of bialgebras as Ext in certain category of «tetramodules» &c (see Shoikhet's arXiv:0907.3335 for details).
May
5
comment $p$ is an odd prime number where $p=3k+1\Longleftrightarrow\exists a,b\in\Bbb Z^+$ such that $p=a^2+ab+b^2$
Related: Fermat's Christmas theorem on sums of two squares...
May
5
comment $p$ is an odd prime number where $p=3k+1\Longleftrightarrow\exists a,b\in\Bbb Z^+$ such that $p=a^2+ab+b^2$
...so as an answer this is substantially incomplete, and as a hint — almost misleading.
May
5
comment $p$ is an odd prime number where $p=3k+1\Longleftrightarrow\exists a,b\in\Bbb Z^+$ such that $p=a^2+ab+b^2$
If you know e.g. a proof that $p=a^2+b^2$ iff $p=4k+1$ you can try to do smth analogous. If not — perhaps you should try to use some number theory text book (e.g. Ireland–Rosen).