/sci/ - Science & Math

>>11766923
hmmm, interesting, thanks! I guess my only other question is if the sequence [math] \int_a^bf\ d\alpha_k [/math] would converge under similar conditions, or if not Riemann integrable implies that the sequence wouldn't converge

Nice, I'll have two of those down by the end of my undergrad

Who needs sleep when you finally crack the problem you have been struggling with?

Yes, this is correct.

>>9245586
>Both ways seem like crude fixes but at this point I'm pretty sure there isn't any other more straightforward way (I've looked through a bunch of textbooks and as soon as they reach the topic of the conjugation action of a group onto itself they quietly abandon category theory and do things in a handwavy way that mixes groups (not as tuples) and sets.
Yes, I have encountered the same.

>I'm honestly not sure what you mean here.
I was meaning something like this: if a group [math]G[/math] acts on some object [math]X[/math] in a category, then the action could maybe be expressed as a function [math]\varphi_g \colon \text{Hom}(X, X)\to\text{Hom}(X, X)[/math].

>This has me more bewildered. I've only briefly studied mobious transformations and wasn't aware they formed a group. Moreover you're talking about homeomorphisms so I guess you're talking about functors from that group to the category of topological spaces?
In hyperbolic geometry, you have this group [math]\mathbf{Möb}(\mathbb{H})=\{ z\mapsto \frac{az + b}{cz + d}\ |\ ad -bc >0\}[/math], and this is indeed a group. It's action is quite easy: [math]\gamma .x = \gamma(x)[/math], but this can be thought of as [math]1_{\mathbb{H}} \mapsto \gamma[/math], or actually just the composition of the Möbius transfromations from the left if I remember correctly. This is just one example, but it's good to have something concrete to motivate the abstract.