The h Bar

Let $\mathcal H'(M)$ be the space of functions spanned by the set of harmonic functions $f$ that have the property that when restricted to each unbounded component of $M-D$ it is bounded either from above or from below for some compact subset $D\subset M$.

Slereah

For the line with two origins, the line that passes through $0$ and the line that passes through $0^*$ are two valid solutions

0celouvskyopoulo7

are they?

have you actually proved that

I'm thinking there's some issue with the notion of curve in such a space

Slereah

Well it's certainly the case if we consider the equation on each of the two coordinate patches

The issue might be related to the non-unicity of limits

0celouvskyopoulo7

9:12 PM

I thought you couldn't have metrics or somthing

@Slereah aha

that's getting closer to the issue

Slereah

Yeah you have sequences with two limits

$$\lim_{n \to \infty} \frac{1}{n} = 0$$ $$\lim_{n \to \infty} \frac{1}{n} = 0^*$$

both valid limits

0celouvskyopoulo7

So I'm not sure that a curve even makes sense in that space

GPhys

our quantum professor was sick and I just got back the results of a chest xray confirming I have pneumonia

Slereah

Well it makes sense, but they may get weird

0celouvskyopoulo7

our notion of continuity makes no sense

GPhys

9:14 PM

when can I fairly say my quantum homework is killing me

Slereah

yeah that's why I'm having big doubts about how to define vector fields on 'em

Since that involves derivative and that may get delicate

Phase

wait, do I lose rep for downvoting things?

Slereah

The line with two origins and the splitting real lines aren't too bad, but then you have weird manifolds like the complete feather, where every point has a branching point

GPhys

@Phase answers, yes

Slereah

How do you define anything there

Phase

9:15 PM

@GPhys oh geez

0celouvskyopoulo7

You need to use the ol' $f^{-1}$ definition of continuity.

Good luck.

GPhys

of course I could answer questions on physics.SE, but I will quickly relose the rep downvoting the endless swath of garbage answers

Slereah

yeah, quite so

hopefully derivatives will be unique and be the same on all patches

0celouvskyopoulo7

There should be a rule that when a post has -2, you get the rep loss refunded

Slereah

Otherwise, I'm not sure what happens

0celouvskyopoulo7

9:17 PM

So you lose rep for -1 on random posts

But if someone agrees with you then it's likely shit

GPhys

at least some answers are so bad that I can just report them instead of wasting vote rep physics.stackexchange.com/a/329470/10908

Slereah

the only thing even close to calculus I found on the topic is that for a non-Hausdorff manifold, if $p$ and $q$ are adjacent point, then $f(p) = f(q)$ for a continuous function

0celouvskyopoulo7

sure

I believe it

hence they're terrible

why are you doing this to yourself?

Slereah

it is an intriguing topic

non-paracompact manifolds less so, for some reason

I guess because they're less easy to visualize

0celouvskyopoulo7

...fundamental theorem of calculus?

Slereah

9:26 PM

It does not seem fairly obvious

0celouvskyopoulo7

16=4^2

is that the hint?

This is so cryptically written

Where the hell did the $4$ come from anyway lol

Slereah

David Gauld is the guy who's way into non-paracompact manifolds

He loves everything long

0celouvskyopoulo7

I mean, TFC is $f(x)=\int_0^L \gamma'(t)\cdot\nabla f\,dt$, where $L=r(p,x)$, $x\in B_p(4)$ and $\gamma$ is the geodesic from $p$ to $x$.

Ah yes

FTC

Crazy!

Slereah

why the 16 though

oh, i guess it is $4^2$

0celouvskyopoulo7

9:43 PM

yeah $L$ is at most $4$

So that integral is $\le 4\sup |\nabla f|$

Slereah

And I suppose that $\gamma'$ is always of norm $1$

0celouvskyopoulo7

Yeah I'm picking arc length param

Slereah

Hm

I should try to find Krasnikov's email again

he was last seen working at the observatory of Pulvoko

In Mother Russia

he's not listed on the staff page

0celouvskyopoulo7

Is he alive?

Slereah

his last paper was in 2016

ah, found an email

S.V.Krasnikov@mail.ru

This one seems more modern as it's not in the soviet union

Let's try again

In that paper there's $B \star C \in \mathrm{Bd} N_1$

Slereah

10:06 PM

It's definately a function that outputs a manifold point

Mail is sent, hopefully it's not dead

it's slightly annoying that Carroll, HE and Wald all use a different version of the Riemann tensor

gotta keep track of what indices are what

0celouvskyopoulo7

...what?

Oh

There's also two sign conventions

be careful

I think Wald has $G=-T$, no?

Weinberg has $G=-T$

Slereah

Oh bother

Slereah

10:41 PM

Wald says that the trace of the EFE gives $R = T$, but what happens to the Ricci scalar term?

Isn't the trace of the metric tensor $g_{\mu\nu} g^{\mu\nu} = n$?

Or the signature or something

0celouvskyopoulo7

halp

set theory

what is $(M-A)\cap U$

@Slereah it's 4

So $R_{ab}-\frac{1}{2}Rg_{ab}\mapsto R-\frac{4}{2}R=-R$

Slereah

Ah, I see

But of course that's only valid in 4 dimensions

0celouvskyopoulo7

spacetime is 4-dimensional

Slereah

Says you

I think that would be $(M \cap U) \setminus (M \cap U)$

0celouvskyopoulo7

that's $\emptyset$

Slereah

10:48 PM

Algebra of sets

The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union, the meet operator being intersection, the complement operator being set complement, the bottom being ∅ {\displaystyle...

$$(B\cap C)\setminus A=B\cap (C\setminus A)$$

0celouvskyopoulo7

Set Theory is too hard

Slereah

Set algebra is like the easiest part

you can just draw circles to figure it out

0celouvskyopoulo7

mb

Slereah

The latex glossary works like shit

I don't know how I managed to work it out during my last thesis

I think I'm gonna use the nomencl package instead

Cows

11:20 PM

@0celouvskyopoulo7 I just read the monotone class stuff. "We
leave it as an exercise to the reader to show that **S** is a monotone class that contains **A**. I suppose I will be writing a proof. hehe. I should be able to do this.

Transcript for