The h Bar

Slereah

20:04

First ep of the Sopranos is downloaded

Let's see

DanielSank

I'm so confused.

I'm still "vaguely around"... what?

"cheater hookup"?

DMac

@EmilioPisanty, I accidentally removed your answer from my question by accepting "move to chat" please re-add if you want. I'm going to delete the question though because it is unclear unless you think it should stay. I'm at work & so I can't stick around to see a response here. Thanks for your thoughtful answer.

Emilio Pisanty

@DMac No. I personally deleted my answer because the debate was not going anywhere and I was not interested in repeating myself.

You can never remove other people's content on Stack Exchange unless (i) you delete your question and there exists a single answer with no upvotes, which gets deleted with the question, (ii) you're a moderator, or (iii) you vote to delete other content as part of a quorum of at least three different users (full rules here).

@DanielSank cf.

37 mins ago, by Emilio Pisanty

Writing proposals and justifications is so much easier when you work on superconductivity. It's like activating a science cheat code.

skullpatrol

20:22

we were pretty much eliminated from the playoffs last night :'(

some nice hats here

Anonymous

20:39

In this paper, I'm a bit confused about this stuff: $\chi(p)$ (so-called-Euler number) has been said to be $N_B(p)-N_W(p)$ (which they claim is a result from Topology), $p$ being the probability of a pixel/point being "black". $N_B(p)$ is the number of black clusters given a certain $p$. And similarly $N_W(p)$ is the number of white clusters given the same probability $p$.

Anonymous

Now, I've looked up Euler's characteristic $\chi$ from Wikipedia, but it doesn't mention any formula similar to $\chi (p)=N_B(p)-N_W(p)$.

Anonymous

So, my question is: What is the significance of the Euler's number $\chi(p)=N_B(p)-N_A(p)$ in Topology and what are it's applications? Some book or source discussing this would be helpful. The paper

Anonymous

which they reference is another similar paper which doesn't explain anything about from where this Euler's number originates. They just say it is a "topological measure". This is so weird...Is $\chi(p)=N_B(p)-N_A(p)$ a legit thing in Topology?

DanielSank

@EmilioPisanty bah

Anonymous

@BalarkaSen Halp needed

Anonymous

20:50

Oh, hey you're there :P Hi

Anonymous

Any idea about that euler number thingy?

Balarka Sen

I am busy right now. If you can give me the context maybe I can help

I don't understand your definition of $\chi$

0celo7

@Slereah apparently the first episode was a comedy

Balarka Sen

and I don't really want to read pages of stuff right now

Slereah

@0celo7 i can see why they went with drama

I didn't realize it was a comedy

0celo7

20:53

@Slereah it worked out for them

Slereah

which is a bad sign for a comedy

Anonymous

@BalarkaSen It's not my definition, actually. I found it on that paper. And can't relate it with the Euler's characteristic given in wikipedia (which is general one used in Topology)

Slereah

I'm trying to finish that electronics thing but the only thing in mypart box for a switch is a weird DPDT switch

No idea how it works

Balarka Sen

Yes I know it's not your definition. I'm asking you to give a tl;dr of the paper's definition

I don't want to read the paper

Anonymous

Alright, lemme try to summarize it:

Anonymous

20:56

Anonymous

Say we have a 2D plane containing $N \times N$ pixels

Anonymous

Then, suppose we have a probability $p$, of a pixel being black.

Balarka Sen

ok

Anonymous

Then, look at the figure (a) for example. If you use randomization algorithm, and set $p=0.17$

Anonymous

Randomly black pixels pop up in the white "sea"

Anonymous

20:59

And you get something that looks like 2(a)

Balarka Sen

right

Anonymous

Now, some of these black pixels may pop up adjacently

Anonymous

And form tiny black clusters

Anonymous

Of more than 1-pixel

Balarka Sen

so N_B(p) is # of black pixels and N_W(p) is # of white pixels?

Anonymous

21:00

BUT, at $p=0.17$, the white pixels form just 1 cluster (notice 2(a)) i.e. the white sea

Balarka Sen

oh clusters

not pixels

Anonymous

@BalarkaSen Just replace that with "clusters"

Anonymous

Yes

Anonymous

After that I've stated the definition of Euler's number they are using

Anonymous

(which they claim to be a result from Topology)

Balarka Sen

21:02

Ahhh I see what's happening

Clever

heheheh this is topology bro

Anonymous

Phew :P

Anonymous

Explain :P

Balarka Sen

I'm still processing it but the idea is pretty clearly that you're counting components. In 2(a) N_W(0.17) = 1 because the space of white pixels form a connected region on the plane

Anonymous

@BalarkaSen Right

Balarka Sen

Let me think about it for a while. This should be apparent

Anonymous

21:05

Alright, lemme know when you get it :)

Anonymous

They also talk about fractal dimensions and stuff in that paper which I'm confused about. But I'll ask that later.

Slereah

So mister Italian just really like ducks

bolbteppa

Can we all just agree that Kaku is an absolute genius

Anonymous

He's a good football player...

bolbteppa

Football of the future

Slereah

21:16

Phew

finally figured out how that switch works

Who sells an electronic kit with only 6 pin switches

Amma gonna get some two pin switches

Anonymous

Get a 0 pin one

Slereah

Hm

I'd better put on some pants before soldering

2

Or things might turn out bad

Anonymous

lol

vzn

wow, spking of climate chg... Trump Decides Climate Change Is No Longer A National Security Threat / huffpost

Balarka Sen

@Blue say you have a 3x3 thing

first picture: everything except the center square is black

second picture: everything except the center and the below-center square is black

These two seem to have the same $\chi$ fam

So how is this a good invariant?

(Or am I misunderstanding?)

Anonymous

21:30

@BalarkaSen What do you mean by invariant in this context?

Balarka Sen

things which determine if two things are distinct or not

Anonymous

$\chi$ is actually a variable according to the paper

Balarka Sen

determinant of a matrix is an invariant which detects invertibility

@Blue That's not what I meant. Whatever.

Anonymous

Balarka Sen

So do you agree $\chi$ are the same for both cases?

Anonymous

21:33

Wait. In your first picture $\chi=1-1=0$ and in your second case too $\chi=1-1=0$

Anonymous

Okay, yes

Balarka Sen

Okay. That's uninspiring.

The Euler characteristic is meant to be a topological invariant. Those two configurations obviously are not the same

topologically

Anonymous

They are not defining $\chi$ as the invariant factor. They're trying to find an invariant factor which is in the next part of the paper

Anonymous

There's some underlying invariance factor which they're trying to find

Anonymous

See this graph:

Balarka Sen

21:36

if chi is not invariant under topological considerations, there's no way it relates to Euler characteristic

Anonymous

@BalarkaSen Right, I get your point. I'll complain about that tomorrow, to them :P

Anonymous

Thanks though

Anonymous

But just have a look at the bottom left para of the picture

Balarka Sen

The Vogel-Hoffman-Roth paper is more interesting

It says $\int_{\delta X} 1/r ds = 2\pi\chi(X)$

horrid notation, but I think there's a way to decode that

Anonymous

21:40

Heh, yeah. They say it's a topological measure of some sort

Balarka Sen

If you integrate $1/z$ over the unit circle you get $2\pi$

so I don't see how $\chi$ of my first picture is $0$ not $1$

Anonymous

@BalarkaSen Isn't that $2\pi i$ or am I misremembering....

Balarka Sen

er yeah that

i'm normalizing

Anonymous

Okay I see

Balarka Sen

The point is the picture you get from the my first thing is of a circle

And Euler char of a circle is 1

Anonymous

21:43

@BalarkaSen It's 0

Anonymous

Check wiki

Anonymous

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by χ {\displaystyle \chi } (Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. Leonhard Euler, for whom...

Anonymous

Under the Examples section

Balarka Sen

lol whoops

I meant the second picture

Anonymous

@BalarkaSen Which picture?

Anonymous

21:46

16 mins ago, by Balarka Sen

second picture: everything except the center and the below-center square is black

Anonymous

Ah okay

Slereah

switch is soldered, and I only have minor burns

Anonymous

@BalarkaSen I actually didn't understand what you meant by "below center" is black

Balarka Sen

@Blue No, wait up, I'm giving you the wrong examples.

Anonymous

Alrighty

Balarka Sen

21:48

I was thinking of the disk, in which case your thing is indeed $1$.

Anonymous

Right

Anonymous

For disc it is 1

Balarka Sen

But this has to be garbage, because just fill a square of a 2x2 box

$\chi_{garbage} = 1 - 1 = 0$ in that case, but the Euler characteristic of a point is 1

Anonymous

@BalarkaSen You are filling one of the blocks of that box with black?

Balarka Sen

This is measure something other that the topology of the black subset

@Blue Yes

Anonymous

21:51

And how does that relate to a "point"?

Balarka Sen

Because that black square is a point, bro :P

1 min ago, by Balarka Sen

This is measure something other that the topology of the black subset

Anonymous

You mean "This is measure of something other that the topology of the black subset" I think. But okay

Anonymous

I'm getting it a bit

Balarka Sen

@Blue You mean "But okay."

Anonymous

@BalarkaSen I didn't want to annoy you. I thought that is some mathematical term which I don't know.

Balarka Sen

21:55

Oh I wasn't annoyed I was joking

i don't really see how this relates to Euler characteristic

Anonymous

Mhm :/

Balarka Sen

like even if you counted just $N_W$ of $N_B$ that'd be sensible

subtracting one from another is not very sensible

Anonymous

I think they should just stop calling $\chi$ the Euler number, and just say they're trying to find an invariant

Anonymous

which is what they do in the next part of the paper anyway

Balarka Sen

If there're calling it an Euler number surely there's some relation

Anonymous

21:58

@BalarkaSen The references they give at the end of the paper lead to another physics paper which is equally vague :P

Anonymous

Physicists abusing math as usual I guess!!!

Slereah

Soldering wire to pins is not a fun business

Balarka Sen

It's not clear what is the topological object of interest

Anonymous

@BalarkaSen So, I need a bit of advice: It is clear from our discussion that according to their definition of $\chi$, it is not a topological invariant, because in the examples we discussed, even for different arrangements, we find that $\chi$ can be same (say $0$). Also, you mentioned that case for that $2\times 2$ grid. So, should I raise that objection when I meet the authors tomorrow? I don't know much of this stuff other than what I've extrapolated from Wikipedia though...

DanielSank

What's going on in here?

Anonymous

22:11

They told me to work on that paper. So it seems I have to seriously learn the parts of Topology which are related to it

Anonymous

@DanielSank Balarka giving me some Topology advice :P

Anonymous

BTW I also need to ask them what their "topological object of interest" is (if they can clarify...well and good...I'll tell you tomorrow)

Balarka Sen

@Blue Well I definitely think there's some topological relevance to this

It's not very obvious to me what that is

DanielSank

Y'all heard of this?

6

Anonymous

I see. Alright. I'll try asking them tomorrow. The problem is that the first author won't be present. I'll have to ask the co-authors. Anyhow, thanks a ton for the help

Anonymous

22:16

I'll later ask you about topo book recommendations :P

Anonymous

See you

0celo7

Bredon, Hatcher, Munkres

Anything more and you’ve lost yourself

ooolb

22:32

@DanielSank there's barely anything on there though

every link i click leads to a blank article

DanielSank

Yes but it's a good idea.

0celo7

@Slereah so how are the sops?

Slereah

it's aight so far

Cows

22:58

It's official

I have a stand up desk now

I code while standing. . . . LIKE A BOSS

:D

Balarka Sen

no like a boss memes in 2017 pls

Cows

hehehe ok ok lol :P

DanielSank

23:13

@Cows excellent

@BalarkaSen no. I refuse

Emilio Pisanty

@DanielSank I've seen it a couple of times. I think I'm going to wait until it's a good bit more populated with content

I'm not sure of how well the fixed structure can cope with expansion

I do hope they iron that out, though, it looks nice

ooolb

23:32

@DanielSank the idea in itself is worthless

the idea is all in the effort

and it's not special at all, there are lots of websites doing the same thing but even more ambitiously, there is one at my university for example

Cows

for the lolz. Live data visualizations coding, some data science , and sentiment anal

tahkevin Live Stream

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