@anon btw, the definition is simple: $\Delta_\sigma u = d(\sigma d^c u)$, where $\sigma$ is a smooth function which "twists" Laplacian, $d = \partial + \bar \partial$, $d^c = -i (\partial - \bar \partial)$, and $\partial$, $\bar \partial$ are the standard Dolbeault operators.
@JasperLoy I've found better expositors, I guess. Though I still have to read a great deal of it. He has Galois Theory, Modules and some other very interesting stuff.
@Complexanalysis Oh, because the condition $q\nmid p-1$ forces the $q$-Sylow generated by an elt of order $q$ to be normal. The $p$-Sylow generated by an elt of order $p$ is already normal because it has index the minimum prime divisor of $G$.
By coprimality, $\langle a\rangle\cap\langle b\rangle =1$, so $G\simeq \langle a \rangle\times \langle b\rangle$.
One also shows $G=\langle a,b\rangle$ first, though.
Prove the following:
$$\lim_{n \to \infty} \displaystyle \int_0 ^{2\pi} \frac{\sin nx}{x^2 + n^2} dx = 0$$
How would I prove this? I know you have to show your steps, but I'm literally stuck on the first one, so I can't.
I was thinking that if $f$ is continous on $(a,b)$, and g has a period of $T$ then $$ \lim_{n \to \infty} \int_a^b f(x) g(nx)\,\mathrm{d}x = \frac{1}{T} \int_0^T g(x)\,\mathrm{d}x \int_a^b f(x) \,\mathrm{d}x $$
@N3buchadnezzar I find Bezout's Identity, the Euler-Maclaurin Sum Formula, the Central Limit Theorem, Bernoulli's Inequality to be very useful. I don't know about favorites.
@N3buchadnezzar Did you see to what that comment referred?
The homology section of the book uses affine spaces, but I feel that everything that was done so far could have been done using vector spaces. Since I am not used to affine spaces as I am used to vector spaces I prefer the vector space viewpoint
Would it be OK if whenever any definition that uses affine spaces I just replace the word with vector spaces
In other words: What is the advantage of using affine spaces instead of vector spaces
@Amr students of vector calculus often ask what's the difference between points and vectors (arrows). the idea is that the points just sit there, inert, while the vectors act on them by translation. geometrically, this is the "correct" way of thinking about things, as e.g. in the real world choices of origins for coordinate systems are artificial and to some extent arbitrary, as are the subspaces through the origin compared to affine subspaces. don't know about homology though.
@Charlie refreshing my memory with wikipedia, it seems to me the idea is that a linear map $\Bbb C^n\to\Bbb C^m$ is comprised of a rotation on the domain $\Bbb C^n$, followed by the projection $\Bbb C^n\to\Bbb C^m$, followed by scaling of the axes (the amount each axis is scaled by corresponds to an entry of the diagonal matrix in the SVD), followed by another rotation in $\Bbb C^m$.
@PedroTamaroff I am working on writing a topology / functional analysis question. It's yucky, but it's ultimately for my algebraic and geometric curiosity.
@robjohn if you have constructed an confidence interval, is there some easy way to find only the upper confidence level, or do u have to do all calculations again?
@Danny The upper confidence level for $95\%$ uses the same computation as the confidence level for $90\%$. For other confidence levels, you need to recompute.
I had this prof in Cincinnati who did kind of a pseudo-Moore thing, where he had students take turns giving the lectures while he asked questions. But we still used a book and used it to prove the main results.
I'm not sure whether that style has a name or not.
Nah, just lazy prof :) it goes slowly, students learn well what they have to lecture on, and not the rest. Students rarely have the perspective to give insights a good prof can.
Prove the following:
$$\lim_{n \to \infty} \displaystyle \int_0 ^{2\pi} \frac{\sin nx}{x^2 + n^2} dx = 0$$
How would I prove this? I know you have to show your steps, but I'm literally stuck on the first one, so I can't.
