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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).
May
5
comment $p$ is an odd prime number where $p=3k+1\Longleftrightarrow\exists a,b\in\Bbb Z^+$ such that $p=a^2+ab+b^2$
The direction '$-3$ is a residue => $p$ is representable' is quite non-trivial and requires either factorization in Eisenstein integers or some clever descent argument.
May
4
comment $\operatorname{Hom}_\Lambda (B, \operatorname{Hom}_{\Bbb Z}(\Lambda, X)) = \operatorname{Hom}_\Bbb Z (B, X)$
Hint: $\operatorname{Hom}_{\mathbb Z}(\Lambda,X)=X[G]$.
May
4
comment A question on the classifying space $BG$, its universal property (?), and the stack $[\bullet/G]$
If $G$ is a group scheme, what is $EG$ and $BG$? There are two problems here: (1) you want $EG$ to be contractible — and it's not clear what it should mean for schemes; (2) in all interesting cases $EG$ is infinite-dimensional.
May
4
comment A question on the classifying space $BG$, its universal property (?), and the stack $[\bullet/G]$
And what do you mean by 'the couple ... represents this functor'? Functor can be represented by an object of the category. Yes, $BG$ represents $F_G$.