Once again, I am mystified by a statement in Reed & Simon.
After proving that the compactness of a space is equivalent to every net having a convergent subnet (a generalization of the usual Bolzano-Weierstrass theorem on $\mathbb{R}$, if I'm not mistaken), they state the following:
Example 1: The unit ball in $\ell_2$ is not compact in the metric topology. No subset of a sequence of orthonormal elements can converge
Is there any way anyone can enlighten me as to what this means?
@Danu Hmm...since all finite-dimensional vector spaces of the same dimension are isomorphic, and the unit ball in $\mathbb{R}^n$ is compact, the unit ball in all finite-dimensional vector spaces will be compact
It is clear at least that the basis element argument cannot work with finitely many dimensions because it ends up being cyclic and even if you 'perturb' it so it doesn't actually repeat itself you do end up getting infinitely many infinitely close elements of the sequence
oh yeah
Sorry - I still associate isomorphisms with iso-metries which are, I think, more difficult generally
but a bicontinuous bijection isn't that difficult, I guess (to set up)
Yeah, but even for the infinite-dimensional ones, the Hilbert spaces at least are also relatively boring: If they have a basis indexed by a set $B$, then they are isometrically isomorphic to the space of sequences $l^2(B)$ (see Wiki).
Not that I don't absolutely love the both of you, <3, but this kind of chat does seem better suited for a separate chat room given that it's been going on for weeks
@Jim If you think it shouldn't be here... I honestly prefer having some chat going on rather than just silence (see also my extended discussions on chess with @Phonon)
The claim In order to track the evolution of the electron distribution in the downstream we follow the method of <someone>, where the electrons are assumed frozen to the flow (i.e. diffusion is neglected)
I'm not sure I buy it right now. I'm going to have to go and read <someone>'s paper
$\partial_t\psi+\mathbf u\cdot\nabla\psi = \kappa\nabla^2\psi+p\left(\nabla\cdot\mathbf u\right)\partial_p\psi$ is the general transport equation used in diffusive shock acceleration
They took $\kappa=0$, basically
@ACuriousMind Diffusive shock acceleration is the mathematical model to the physical model of Fermi acceleration
It's more of DSA is FA but FA is not necessarily DSA type relation
Hmm, it seems like the argument is that since SNR 1987 is still in the free expansion phase, the ejecta is being pushed faster & further than the diffusion rate. That it's not until the remnant ages into the Sedov phase that the diffusion becomes necessary.
@Phonon Why would you think I forgot to apply? Don't worry, all is well (though, due to some technicalities, I might only formally be studying as a master in the coming summer term)
I'm currently unoccupied by academic matters though - sweet, free time :)
@Phonon The dreaded lab courses we have to take have a idiotic booking system that prevented my team from booking and completing all the required ones this semester
And since these are required, you don't get your BSc until they are done
But it's not really a problem, they do not care what you are studying, you can visit (and take exams) in the master courses no matter what.
@Danu If only we had been to fault for breaking something, but it happened twice that we arrived, worked for some hours only to find out it had been broken all along
My penchant for theory may in not a small part be influenced by the terrible organisation of these courses.
On an unrelated note: I second @Jim in asking Is this a joke?.
@ACuriousMind sry was afk for a bit, ah ok I see then, c'mon guys that's nothing...I have 5 hours of lab per week LOL (starting with transmission electron microscopy...)
but good to hear you're still admissible to exams and all
They were first year physics students, some were 2nd or 3rd year
But this was halfway through the year
"Sir, my oscilloscope isn't showing me anything" "Is it turned on?" "No, but I thought the measurement powered it" "Yes, that's why they provide a giant, useless ON button and a power cord"
It didn't happen with oscilloscopes, but I have to admit that I have overlooked more or less obvious buttons like "ON" or "REPEAT" or something like that multiple times
Something about studying does not prepare you for noticing buttons, it seems
What is the name of the phenomenon where you illuminate a film at varying incidence angles, until at some angle the light couples and makes surface plasmons on the film?
Translation: I'm really desperate to give this person undeserved answers to their homework, but the broken system is preventing me. Someone should fix this.
@ChrisWhite If they're talking about a faculty position at a PhD granting institute, that might be true. If they're talking about any university, they're probably wrong