/sci/ - Science & Math

>>11388589
No. We just need the Fourier coefficients to be defined, then [math]\mathcal{F}(f-a_ne_n)_m = \langle f,e_m\rangle - a_n \langle e_m,e_n\rangle = a_m - a_n \delta_{nm}[/math] which gives [math]0[/math] when [math]n=m[/math]. Repeat for finitely many elements and you have [math]\mathcal{F}\left(f-\sum_{|n|\leq N}a_ne_n\right)_m = 0[/math] for all [math]m \leq N[/math] so [math]f - \sum_{|n|\leq N}a_ne_n[/math] is orthogonal to [math]\operatorname{Span}_{|n|<N}\{e_n\}[/math]. We have not assumed any convergence of the partial sum or if [math]\sum_n \mathcal{F}(f)_n e_n \rightarrow f[/math] uniformly.

>>11101892
Sorry for trying to facilitate sincere mathematical discussions.

>>10176230
The quantum state [math]|X\rangle[/math] merges with the state [math]|\Psi\rangle[/math] of the blackhole and the quantum state [math]|Y\rangle[/math] is entangled with [math]|\Psi\rangle[/math].
Now the problem comes from the fact that blackholes evaporate via Hawking radiation. What happens if we keep feeding entangled particles into the blackhole so it evaporates away to the vacuum [math]|0\rangle[/math]? There's nothing in the [math]|0\rangle[/math] for [math]|Y\rangle[/math] outside of the event horizon to entangle with, so is the entanglement broken? If this were the case then we lose information about the entangled state [math]|Psi,Y\rangle[/math] we started with, since we now only have [math]|Y\rangle[/math]. This is the blackhole information paradox.
Here's a possible resolution https://arxiv.org/abs/1703.03042

>>8964788
>the thing they give you when you have the best grade
The word is "distinction".

Do more research and get a paper published, and try to sound passionate during interviews. If you don't have the grades at least have the research skills and drive to back it up.
Also I hope your grades increased throughout undergrad, or you're fucked.