/sci/ - Science & Math

>>15093305
i feel genuine pity for you, but am also amazed that you managed to overcome your urge to eat crayons for long enough to type that post

>>11504202
Oops, guess I forgot non-linear Schroedinger problems are too abstract. Sorry.
>their rate of change bound
Yes, then you (formally) take [math]\partial f = \sup \partial x^*[/math] of the subdifferentials in the supremum of the [math]E^*[/math] norm. Use this to study the variations of [math]f[/math] like you would in finite-dimensional calculus on manifolds.

>>11490811
>what does it say about the universe in general
It suggests that ground states might be topologically degenerate.
>What are the anomalies in reality?
Cosmic string in cosmology, monopoles and the AHE/QHE in condensed matter, etc.
>Explain without using formulas ou technical terms please
Think of fields as a big bed sheet and anomalies are the clumps of cum your bf left on it.

>>11484776
>finite linear combination
They're called formal* linear combinations, and endows a specific topology on the space; namely the topology inherited from the direct limit [math]\lim\limits_{\rightarrow} V[/math] of finite-dim vector spaces. This is a specific choice of topology, not sure why you thought you don't need any for this in the first place.
>>11484780
[math][g,N] =1[/math] it means they commute, so [math]gNg^{-1}=N[/math] for every [math]g\in H[/math].
>Also, how does
Subgroups [math]N\subset H[/math] are normal iff [math]N\in \operatorname{ker}h[/math]. First isomorphism theorem then gets you [math]G \cong \operatorname{im}h/\operatorname{ker}h = S_X/\operatorname{ker}h[/math]. If there are no such normals then [math]\operatorname{ker}h=1[/math] so [math]G \cong S_X[/math].

>>11470940
Read Tinkham or de Gennes hun.

>>11389028
Why would it? All quantum corrections do is to impart uncertainty and noise to the [math]exact[/math] position of the particle, the classical contribution (consistent with Einstein/Newton/Kepler) is still the most significant, especially at large scales.

>>11377856
>it's constant on the connected components
And? You do know that any connected open/closed has connected preimage, right?
>>11377862
>If they don't agree on the intersection the union isn't a function
No shit? The point is that even if that's the case it can [math]still[/math] not be a function. See monopole harmonics.

>>11265112
>nice elaboration of QED on a grid
Lattice QFT and is in general not a nice way of doing foundational QFT. There's a reason why C/AQFT and quantum jets start by quantizing the fields from the get-go, and the reason is that the continuum limit fucks with the Hilbert space of states.

7: If [math]X_n \rightarrow X[/math] is an inductive limit and [math]l:X^*\rightarrow \mathbb{C}[/math] is a functional such that (some technical conditions) then [math]l[/math] is continuous iff [math]l|_{X_n}[/math] is each continuous.

>>10968767
https://en.wikipedia.org/wiki/Mackey_topology

>>10936278
>no non-vanishing vector field.
No non-vanishing [math]tangent[/math] vector field.

>>10558132
The post being riddled with typos and grammatical/jargon-related errors aside:
>Tbe word "tensor field" isnt present in any popular relativity book
Read Nakahara, since you probably doesn't know any more diff top beyond the typical calc material.
>differential map given for a smooth function
The differential map [math]d: \Omega^\ast \rightarrow \Omega^{\ast+1}[/math] is given at the level of forms, not just smooth functions. It needs to satisfy [math]d^2 = 0[/math] which requires a lot more structure than just the smooth functions since [math]d[/math] shifts the degree.
>between manifolds is the same as the jacobian which can be defined without any appeal to the charts.
Absolutely wrong on both accounts. The Jacobian [math]matrix[/math] (not its coordinate free version, [math]\phi^*(dx)[/math] which does NOT require Frechet in the slightest) depends crucially on the coordinate functions [math]x_i :M \mapsto U [/math] coming from the choice of a chart. This is literally where the matrix structure came from.
Learn diff geo before posting please.
>it literally doesnt matter
>but I pick the identity as the chart
You do know for [math]n=4[/math] there are uncountably many inequivalent smooth structures on [math]\mathbb{R}^4[/math] aside from the identity right? Literally every single one of these give rise to non-diffeomorphic charts for which the Jacobians (or any form for that matter) aren't smoothly homotopic over [math]\mathbb{R}^4[/math].
Fuck's sake kid, stop embarrassing yourself.

>>10489889
>very related to Stoke's.
It isn't. The hairy ball theorem is related to the non-parallelizability of [math]S^2[/math], which is proven with topological K-theory, not de Rham cohomology.

>>10331940
Why is this the only thing he's done? And it's literally just about using the chiral anomaly to symmetry-break Dirac fermions into Weyl ones; any highschooler should be able to do this.

>>10225798
>It makes predictions that are experimentally verified
Yeah until the ~130th term in the perturbative expansion right? After which the QED S-matrix diverges without bound. And darn where's that chiral Goldstone boson in QCD??? It must be here somewhere because who [math]cares[/math] about rigour?? Right??? xddXDDdxd

>now we induct on n
>backwards

>>10106547
>keeping healthy makes your brain more efficient
Wow who would've thought??

>>9402556
>No, I meant mixed there.
No I'm pretty sure you switched them around everywhere. By assuming [math]I(x) = |f_1(x)+f_2(x)|^2[/math] is the probability density it's already implying that [math](f_1(x),f_2(x))[/math] is in a pure state (with a diagonal density matrix).
>I'm literally saying they are nonsensical and are a reason why classical probability theory fails to explain quantum phenomena.
And this is wrong because classical probability is still the foundation of QM. As I've said, a result that violates a well-established theory/statement must be due to our lack of proper understanding of it, not because of the fault of the theory. I (or anyone I know) would not say probability theory, classical or otherwise, fails to describe quantum mechanics in the same way that I wouldn't say the AB effect undermines gauge invariance.
Anyways this is now leaning towards philosophy so I'll stop replying if you're gonna continue down this road.

>>9058842
So your only argument is that KanColle/Idolshit is only relevant to recent cultural trends in Japan while Touhou goes all the way back to the Warring States Era. Glad we agree.
>>9058864
Don't butt into other people's conversations. It's very rude.

>>9027028
>notice how the greek letters connect
Like how the physicists did back in 1920?
Coordinates in the dual basis can be written as contravariant indices so that [math](u^*)^\beta u_\alpha = \delta_\alpha^\beta[/math]. The bra-ket notation is also another example of good notation: the kets [math] |\psi\rangle \in \mathcal{H}[/math] are elements of a Hilbert space and bras [math]\langle \phi | \in \mathcal{H}^*[/math] is an element in the dual, so that the pairing ca be written as [math]\langle\phi |\psi \rangle = \phi(\psi) \in \mathbb{C}[/math].

>>9020883
>probably an adjective
>is wrong
Whoa... so this is the power of mathematical (((intuition)))...

>>9010157
>>9010240
>go on math/physics stackexchange
>"how do I show X is Y?"
>long unreadably verbose post that proves X is Y for generic non-Hausforff non-commutative X and non-compact Dedekind-infinite Lindelof Y as top answer
>"here's a 500 page PDF that talks about X for 5 pages and I'm not going to give you the page number"
>"just look at this one other question that has a link to another question as top answer rofl"
>"if X is Y then [blah]"
>"try reddit lmao"

>>8975504
>oh boo hoo information isn't spoonfed into my facehole
Don't bother, you won't understand it anyway.

>>8958924
>>8958948
It's [math]\delta_{n_1 n_2}[/math].
>>8958975
Use trig identity.