/sci/ - Science & Math

First for having your paper accepted into PRB

>>11560661
>There has to be a way to distinct a plane from a sphere
A plane is contractible while a sphere isn't, this is detected by the Euler character: [math]\chi(\mathbb{R}^n) = -1[/math] while [math]\chi(S^n) = -2\delta_{0,n\mod 2}[/math]. In 2D, by Gauss-Bonnet [math]\chi(X) = \int_X K[/math] where [math]K[/math] is the principal curvature of the tangent bundle [math]TX[/math]; this was the main starting point of Thurston's classification of compact closed 2-folds.

>>11479482
It's not my fault ok, the 4chan TeX fucks up if there are too many [math]math[/math] environments in one line.
>>11479956
Your advisor is a physicist, of course he'd want you to focus on the physics instead of the math. Once you've had enough experience you'll find that focusing too much on the formalism in fact hinders you from making progress.

>>11464366
Ah yeah that part is wrong, my mistake. It just means that [math]\theta[/math] doesn't move but also the function isn't defined on [math]2\pi[/math] anyway while [math]r \rightarrow 0[/math] so the image gets suppressed. Point still stands though, that the image doesn't blow up.

>>11440475
>proof of Baire's category theorem doesn't use category theory

>>11104245
Really? Let's ask Edward Witten, Lisa Jeffrey, Dan Freed and Pavel Etingof about that.

>>10379865
Depends on the lube you're using. Water based should be fine but oil based lube might erode the silicone.
Don't do this everyday though.

>>10270480
Because the prostate is up there. Poop can be easily cleaned out.

>>10188122
Notice that a 1-forms on [math]\mathbb{R}^2[/math] is closed iff its components satisfy the Cauchy-Riemann equations, hence exterior derivative on 1-forms is equivalent to the complex derivative. Since [math]d[/math] manifests as the rot operator (the 2D analogue of the 3D curl operator) on 1-forms, the complex derivative is exactly rot. The fact that holomorphicity is equivalent to form-closure is used to prove the [math]\partial[/math]-Poincare lemma in Dolbeault complexes.

>thought I failed my oral quals
>turned out I didn't

>>10164019
This is basically just computing the expectation of the Casimir element [math]a^\dagger_i a_i[/math] of the Heisenberg algebra in the [math]L^2(X,\mathbb{C})[/math] representation, where [math]X[/math] is the spacetime inside/outside the plates.The discrepancies between the particle numbers for the spacetimes between and outside of the plates should tell you that there's a pressure force being exerted.
>>10164261
Use Heine-Borel. Closed and bounded subsets [math]A,B \subset\mathbb{R}^2[/math] are compact, and since [math]d:\mathbb{R}^2 \times\mathbb{R}^2\rightarrow \mathbb{R}[/math] is continuous [math]d(A,B)[/math] is compact and so it's also closed and bounded.
>>10164870
Use [eqn]\int_\mathbb{C}dz f(z)\csc \pi z = \sum_n f(n)[/eqn]. No need for Parseval or Fourier.

>>10151772
These [math]W_m[/math] are Verma modules. Note that we're trying to represent the [math]affine[/math] Lie algebra of [math]\mathfrak{sl}(2)[/math], not [math]\mathfrak{sl}(2)[/math] itself.

>>10134931
In that case you'd be correct. If [math]W_1 \cap W_2 = 0[/math] then their bases don't intersect (trivial exercise), and [math]L(W_1)\oplus L(W_2)[/math] are just block diagonal matrices [math]W_1^* \otimes W_1 \oplus W_2^* \otimes W_2[/math]. Not much to prove here really.
>>10135245
5000 hours in MS paint.

>>10118217
Yes. [math]G[/math] is absolutely continuous in fact.