/sci/ - Science & Math

>>11558392
It is enough for [math]f[/math] to saturate the Poincare inequality [math]|f|_p \leq |\nabla f|_p[/math]. These functions are discussed briefly here https://arxiv.org/abs/1202.2071.. This might be stronger than what you need however.

>>11486589
>General
Don't.

>>11405895
You're glossing over a [math]lot[/math] of the necessary structures but good effort nevertheless.

>TAing QM2
>talking about normalizability
>state quickly how [math]\psi\in L^1 \implies \psi \in C^\infty \cap L^\infty[/math] only in the Riemann sense and that there are Lebesgue [math]L^1[/math] functions not in [math]L^\infty[/math]
>some kid pipes up "when will we ever encounter this kind of stuff any way"
>go on a whole tangent about Dirac combs and contact potentials
>entire class hates his guts
>>11402035
You do realize Schwarz's original paper on theory of distributions called them "generalized functions" right?

>>11389018
>idiot thinks I was referring to an author
>idiot doesn't know how to google
>is an idiot
https://en.wikipedia.org/wiki/Mackey_topology
chEcKs oUT

>be in class taught by my supervisor
>he's finishing up a proof
>notices that he didn't use a minor hypothesis in the proposition
>look over his proof numerous times and can't find where the hypothesis was used
>I point it out, he lights up and smiles
>"Nice anon, you saved the proof! Now if only you did that on our last paper half as quickly"
>whole class erupts into laughter
>mfw

>>11283471
no u
>>11283819
In standard GR yes, but you can do away with the assumption.x
https://en.wikipedia.org/wiki/Nonsymmetric_gravitational_theory
https://arxiv.org/abs/1512.00207

>>11245959
Write a matrix [math]M \in \mathcal{M}_{n\times n}(R)[/math] as [math]M = \bigoplus_{ij} m_{ij} E_{ij}[/math], where [math]R[/math] is a [math]\mathbb{Z}_{\geq 0}[/math]-module and [math]m_{ij}\in \mathbb{Z}_{\geq 0}[/math], with [math]E_{ij}[/math] the indicator matrix of the [math](i,j)[/math]-th sector. A morphism [math]f = (f_{ij})[/math] maps [math]f_{ij}: m_{ij}E_{ij} \rightarrow k_{ij}E_{ij}[/math] and is hence a module homomorphism [math]R^{m_{ij}} \rightarrow R^{k_{ij}}[/math].

>>10949576
I think "easier" depends on what you want to get at the end. Given the fusion relations as well as an affine representation thereof, you can compute, with TQFT, [math]Z_c(\Sigma_{z_1,\dots,z_m})[/math] on punctured Riemann surfaces, where irreducible characters for the affine reps are inserted at the puncture [math]z \in \Sigma[/math]. This is equivalent to the correlations [math]\langle \phi(z_1),\dots,\phi(z_m)\rangle[/math] on the CFT side, and completely bypasses dealing with highest-weight irreps of the Virasoro primaries. The same goes for the quantum spins (braiding statistics), which you can obtain from Vafa's theorem.

On the other hand, if you want the central charge [math]c[/math] or the anomalous dimension [math]\eta[/math], you need to compute the singular part of the stress-energy tensor [math]\langle T(z)T(w)\rangle \sim \frac{c}{(z-w)^{4-\eta}}[/math]. This is evidently easier with CFT since [math]T[/math] is a bidifferential you can perform contour integrals over, while for TQFT you'll need to assign Hodge line bundles over the moduli space of Riemann surfaces (for which [math]Z[/math] maps into the fibre spaces of these line bundles) and then analytically continue [math]Z[/math] to the compactified moduli, then use factorization to extract the singular part via [math]T \sim \partial Z[/math]. At this point you might as well just be doing CFT.

>>10160632
>The first contact with manifolds.
Well yeah but that's not really the point of contention.
>subsets of [math]\mathbb{R}^n[/math]
Well you can embed any smooth manifold in [math]\mathbb{R}^n[/math] for sufficiently large enough [math]n[/math] but that won't help with visualizing the manifold on a page of a book now would it? The point is that how manifolds are represented on a page does not fully capture its smooth/topological content.
>Differential geometry is ugly anyway.
At least diff geo is local and allows much more freedom in what you're studying. Besides, we have gadgets like spin-structures and principal bundles etc. that are much more powerful than just pretending we're doing vector calc on local charts.

>>9056709
The theory of topological phase transitions developed by Kosterlitz and Thouless in the 1970's was only recognized by Nobel in 2016, and the gap between theoretical and experimental discoveries are only going to widen. The relatively recent idea of non-Abelian excitations might not even be experimentally verified in my lifetime.
Even without all the /x/-tier shit like string theory it still takes a very long time for experimentalists to catch up.

>>9027960
>>9027940
I was born in Taiwan.

>>9018877
>No, it gives you actual proofs.
Interesting, so it's sort of like automated theorem proving in Euclidean geometry via Groebner bases, but weaker.
Can't imagine why mathematicians would try to make themselves jobless though.

https://www.youtube.com/watch?v=m6rfpQXzXu0
>yfw the trend of physics driving new mathematical research is only going to increase

>>8989012
But physicists are solving problems in math now.
https://arxiv.org/abs/1608.03679
>math is too hard for mathematicians
>physics is too hard for physicists
What does this mean?