Mathematics

2:40 AM

Why is this statement false? The "line separation property" asserts that a line has two sides.

It asserts that a point splits a line into two rays, right? That if A, B, and C are colinear, with B between the other two, then the line through those points is the union of the rays BA and BC.

Yes

Not sure exactly how I would interpret the statement "a line has two sides"

My first instinct was that it meant that a line separates a plane into two pieces, or something like that

But the plane separation axiom does define sideness?

Does it? OK, I admit, I don't really know this stuff, sorry. Looking it up now

Ah, I see. So, then, it's the plane separation axiom that says a line has two sides, not the line separation property. And that's why the statement is false. I think.

@notorious (This statement)

2:50 AM

Oh, yes of course. That makes sense. Thank you!

I'm not sure, though

Steven Stewart-Gallus

I was trying to wrap up my head around the calculus of variations and apparently one can define a gradient operator for vector spaces like so? $\left(\nabla_x f\left(x\right)\right) \cdot v =\left. \frac{\mathrm{d} f\left(x + \varepsilon v\right)}{\mathrm{d} \varepsilon} \right\rvert_{\varepsilon = 0}$ I'm not sure this makes sense to me.

Puzzled417

4:10 AM

hi

user147690

4:36 AM

What is 'the cocycle condition'?

user147690

Context: Line bundles on complex manifolds

Mike Miller

4:47 AM

$g_{ab}g_{bc}=g_{ac}$

Surdz

6:07 AM

can I get a confirmation: math.stackexchange.com/questions/1806601/…

6:31 AM

@MikeMiller How does that actually relate to cocycles from cohomology?

Surdz

no idea

Jeff Teoh

What's logarithms used for?

user147690

@Danu I couldn't work out how that answered my question, but was hoping more reading would deduce what any of those letters mean

user147690

I think they are transition functions

Yeah

But I want to know why that's called the cocycle condition. I guess I'll google around a bit.

user147690

6:43 AM

@Danu Should be in Hatcher?

Wiki leads me here

user147690

I tried to read from Hatcher page 198 and made no progress

The subsection "cocycle" has the thing I was looking for.

You could have a look at the book on Riemann surfaces by Forster; the first part of the second chapter. It only deals with Riem. surf. (hence 1D, connected), but it's quite elementary in its wording and stuff.

Maybe you can get a feel for the basics from there.

user147690

That's actually where I am now haha

user147690

Finding my Complex analysis lacking a little

user147690

6:46 AM

(Read a-lot)

I'm taking Forster's course this semester ^^

user147690

Jealous

user147690

Envious*

I'm unable to attend any of the lectures though :( Schedule clashes...

So I'm just attending the exercise classes (I read most of the book over the break, though I wasn't very precise in my reading/skipped technical sections occasionally)

user147690

Now that I know that, you can give me an exposition of the necessary content :D

user147690

6:50 AM

So a Riemann surface is a 2 dimensional manifold that has complex charts for each open subset of the manifold

Holomorphically compatible, yes

user147690

7:10 AM

Do you know where Mike's $g_{ab}$ etc map from and to? @Danu