/sci/ - Science & Math

>>11464416
No you're fine. You're right that you need [math]f[/math] to be continuous on [math]\mathbb{R}^2[/math], which is what I assumed; sorry about the confusion, I thought you were doing the same.
Though your function would've been continuous on the strip but you can make your image do whatever you want there.

>>11447984
Yeah sorry, that's what happens when I try to multitask and do research while at a conference; thanks for your ideas though. I did manage to demonstrate an instance in which Freed-Hopkins's conjecture seems to be true though.
I'll have to think more about how to regularize the trace; for now my only idea is to leverage the finite-dim formula [math]\ln \operatorname{det} = \operatorname{tr}\ln[/math] and [math]\zeta[/math]-regularize, but I'll have to flesh out the details later.

>>11107182
In general [math]\mathbb{R}^n[/math] if something depends only on [math]|x|[/math] then it's symmetric under [math]O(n)[/math], the action of which fixes spheres of radii [math]r = |x|[/math]. These spheres [math]S^{n-1}[/math] can be parameterized with [math]n-1[/math] Euler angles [math]\boldsymbol{\varphi}[/math] (say), hence spherical coordinates in [math]\mathbb{R}^n[/math] can be written as [math](r,\boldsymbol{\varphi})[/math].
Now the Laplacian [math]\Delta[/math] is [math]O(n)[/math]-invariant, i.e. [math][\Delta,O(\boldsymbol{\varphi})] = 0[/math] for all [math]O(\boldsymbol{\varphi})\in O(n)[/math], hence [math]\Delta[/math] is itself independent of [math]\boldsymbol{\varphi}[/math], and only depends only [math]r[/math]. You can then just find the scaling factors in [math]r[/math] from e.g. the first row of the Jacobian [math]J[/math] of the coordinate transform [math]x\mapsto (r,\boldsymbol{\varphi})[/math], and use [math]\partial_i = J_{ir}^{-1}(x)\partial_r[/math] to find the explicit form of [math]\Delta[/math].

>>10305806
Short answer: yes
Long answer: fuck yes