Kogasa

In the context of differential forms, an integral expression isn’t complete without an integral symbol and a differential form to be integrated.

The mathematician also used “operative” instead of, uh, something else, and “associative” instead of “commutative”

I dunno if you’re joking, but yeah there’s IDE plugins that do this. GitHub Copilot grabs context from files in your edit history and you can tell it to edit, refactor, “fix” etc. selections. The more complex actions, the less likely to succeed, though.

“Don’t mention the war”

1. I also have a masters in math and completed all coursework for a PhD. Infinitesimals never came up because they’re not part of standard foundations for analysis. I’d be shocked if they were addressed in any formal capacity in your curriculum, because why would they be? It can be useful to think in terms of infinitesimals for intuition but you should know the difference between intuition and formalism.

2. I didn’t say “infinitesimals don’t have a consistent algebra.” I’m familiar with NSA and other systems admitting infinitesimal-like objects. I said they’re not standard. They aren’t.

3. If you want to use differential forms to define 1D calculus, rather than a NSA/infinitesimal approach, you’ll eventually realize some of your definitions are circular, since differential forms themselves are defined with an implicit understanding of basic calculus. You can get around this circular dependence but only by introducing new definitions that are ultimately less elegant than the standard limit-based ones.

Ok, but no. Infinitesimal-based foundations for calculus aren’t standard and if you try to make this work with differential forms you’ll get a convoluted mess that is far less elegant than the actual definitions. It’s just not founded on actual math. It’s hard for me to argue this with you because it comes down to simply not knowing the definition of a basic concept or having the necessary context to understand why that definition is used instead of others…

No 👍

It doesn’t. Only sometimes it does, because it can be seen as an operator involving a limit of a fraction and sometimes you can commute the limit when the expression is sufficiently regular

The other thing is that it’s legit not a fraction.

It’s a number and complexity refers to functions. The natural inclusion of numbers into functions maps pi to the constant function x -> pi which is O(1).

If you want the time complexity of an algorithm that produces the nth digit of pi, the best known ones are something like O(n log n) with O(1) being impossible.

The direct connection is cool, I just wonder if a P2P connection is actually any better than going through a data center. There’s gonna be intermediate servers right?

Do you need to have Tailscale set up on any network you want to use this on? Because I’m a fan of being able to just throw my domain or IP into any TV and log in

I just use nginx on a tiny Hetzner vps acting as a reverse proxy for my home server. I dunno what the point of Tailscale is here, maybe better latency and fewer network hops in some cases if a p2p connection is possible? But I’ve never had any bandwidth or latency issues doing this

It gets around port forwarding/firewall issues that most people don’t know how to deal with. But putting it behind a paywall kinda kills any chance of it being a benevolent feature.

What specifically constitutes a hole is somewhat ambiguous, but if you pull on the thread a bit, you’ll probably agree that it’s a topological quality and that homotopy groups and homology are good candidates. The most grounded way to approach the topic is with simplicial homology.

Okay

Yeah, it’s a matter of convention rather than opinion really, but among US academia the convention is to exclude 0 from the naturals. I think in France they include it.

Hold on to your hat

http://bettermotherfuckingwebsite.com/

and, you guessed it,

https://thebestmotherfucking.website/

You can imagine tracing a path along a Klein bottle to see that it only has one side. To get more precise than that requires some topological context. If you slice it down the middle it turns into two Möbius strips and an orientation of the Klein bottle would induce an orientation of the strips, which are non-orientable. Alternately it has zero top integer homology, which you can get from looking at a triangulation. The orientable double cover of a Klein bottle is a torus, which is connected (if it were orientable, the double cover would be two disconnected Klein bottles).

Well now we have to find you good Miku songs just in case you are trans.

https://youtu.be/Ljr2wMSBHqU

I mean the specific issue about the binary blobs. Something that might set off alarm bells for you or a security-focused group may not do so for some dude working on a passion project in his free time.