The h Bar

0celo7

5:00 PM

@Danu $\infty\times 0=?$

if we can solve that equation

all will become clear

Slereah

yuggib

gotta go sorry, however that amplituhedron stuff seems not mathematically adequate to me...but may lead to interesting developments

John Rennie

@0celo7 that equation isn't soluble because its meaningless. I take your point though. The task is to rewrite the meaningless equation in a way that is meaningful.

Danu

@0celo7 Semi-related quote by Mukhanov: "Dividing zero by an elliptic operator is like dividing by 5; dividing it by an hyperbolic operator is like... Green's functions!"

0celo7

@JohnRennie that was not my point

ACuriousMind

5:01 PM

@Danu wat

Slereah

ahahah

Danu

@ACuriousMind He really has great intuition :D

John Rennie

@0celo7 and ...

0celo7

@Danu explain please

@JohnRennie and what?

Danu

His point was that about solving $L \varphi=0$ for different types of differential operators $L$

$\varphi=0/L$, obviously

now, the question is: What is $\varphi$?

John Rennie

5:03 PM

@Danu Ah, OK, I don't understand that now :-)

Danu

The quote gives the answer

0celo7

ELI5 why elliptic or hyperbolic matters

ACuriousMind

@Danu The quote seems to say $\varphi = 0$ for elliptic operators, which is false, isn't it?

David Z

our hour is up, it seems... I gotta go, see y'all later

0celo7

bye

Danu

5:04 PM

@ACuriousMind With zero boundary conditions I think it is right?

ACuriousMind

The Poisson equation has non-trivial solutions, unless I am misunderstanding something.

Oh, wrong reply

@Danu uhh... sure

lol

@DavidZ lol

5:05 PM

@0celo7 : you aren't done with me, you're done with physics. You aren't doing physics all day long at all. $\infty\times 0=?$ is trivial math that I've already explained. Not physics.

Danu

@JohnDuffield Can you relax a little?

0celo7

@JohnDuffield If you say so.

Danu

@ACuriousMind Actually I had Mukhanov shout at me one time because of something like this

0celo7

brb changing majors to philosophy

Slereah

But can one really change major

Without changing one's self

deep philosophical thoughts

0celo7

5:06 PM

stfu

you're on absinthe again

ACuriousMind

@Danu Hmm...but you also have Green's functions for the Laplace equation. I don't understand at all what the quote wants to tell me, I guess

Secret

@JohnRennie I rmb it is only somwhere in the beginning of year 2 that I start understand how to exploit symmetry to solve problems in physics, for example

Danu

He had $\Delta\varphi=0$ and asked: So... what is $\varphi$? After he stared at the classroom intensely for a few seconds I hesitantly answered: "...linear?". Mukhanov: "NO, I'M SORRY! ZEEEERO"

Slereah

not all french people are on absinthe

0celo7

true

ACuriousMind

5:07 PM

@Slereah One must first change one's self before one can change one's major.

0celo7

but you are

Slereah

@Danu So what was the answer

ACuriousMind

@Danu ...but, it could be linear! :D

Danu

@Slereah Zero

He implicitly assumed zero boundary conditions

Secret

@JohnRennie I also remember that in my 2 year where I am first being introduced electromagnetism in vector calculus language drives me to a confusion, but then on my 3rd year when doing the electrodynamics courses, suddenly everything clicks

Slereah

5:08 PM

oh

How dickish

0celo7

I know nothing about PDEs either

I really know nothing

Slereah

That's the thing you might want to say explicitely :p

Danu

@0celo7 The sad truth ;)

ACuriousMind

@Danu Well, that's lazy physicists at work again, not stating their assumptions.

0celo7

what are zero boundary conditions

why are they important

Danu

5:09 PM

@ACuriousMind No bad words about Mukhanov from me

He's awesome

Slereah

when phi(x,0) = 0

Danu

@0celo7 Because physical systems are typically localized

ACuriousMind

He probably didn't even specify the function space on which the equation was to be solved, right? :P

0celo7

should I take PDEs or topology next year

Danu

so at infinity you expect to have no fields

John Rennie

5:09 PM

@Secret That's normal. Everyone feels that way. The stuff you found really hard last year feels easy this year. The best way to get good at physics is do lots of physics and keep doing it.

Danu

PDE's first probably

@ACuriousMind Shaddap

0celo7

then I won't ever take topology D:

Danu

@0celo7 ?? Why

0celo7

I'd get 3 semesters of PDEs that way

Danu

You have about 9 years if you wanna take a PhD

Slereah

5:10 PM

topology classes are mostly

drawing potatoes

Danu

I'm sure you'll manage to squeeze it in

0celo7

it's hardly relevant to NE

ACuriousMind

I would take topology. But don't listen to me, I'm just a physicist with an identity crisis because everyone thinks I'm a mathematician :D

0celo7

I'm already planning on taking the 2 semester grad sequence for PDEs

after I've taken analysis, of course

Danu

haha, right

ACuriousMind

5:11 PM

@Slereah The Hawaiian earrings are not potatoes, for instance!

Danu

That's important ;D

Slereah

Topology!

ACuriousMind

(topology examples generally have funny names)

0celo7

actually analysis is a stated prereq

Slereah

:D

0celo7

5:13 PM

@Danu so that will be my fifth year

Secret

T1 is open (forgot term)?
T2 is 2 disjoint regions?
I have no idea about T3 and T4

Slereah

those are the separation axioms

ACuriousMind

@Secret T1-T4 are so-called separation axioms

Slereah

topology is basically what tells you how "near" things are

But not necessarily with values

Danu

Did you guys know they go up to $T_6$, actually? :P

Slereah

5:14 PM

I do

0celo7

topology is like topography, right

Slereah

There's also R

0celo7

that's the impression i got from HE

cool book, that

tells you all about spacetime mountains

ACuriousMind

@Danu There are also things like $T_{3\frac{1}{2}}$, according to the Wiki article :D

Danu

@ACuriousMind Sigh :P

Slereah

5:15 PM

It also tells you how to mess with spacetime

"Just cut some holes"

Danu

Wow, I just disovered something absolutely hilarious

Watching mathematics lectures with YouTube's automatic subtitle functionality

ACuriousMind

@Danu You know how to have fun :P

Secret

(Joke) surely there is no such thing as $T_{\pi}$ right?

PS I don't know much about topology except for regions, holes and connected sums (and the 3 basic topological shapes that work with a connected sum)

Slereah

Do you know about compactness

it's p. important

Danu

@ACuriousMind But really though.

Laughing my ass off at Gromov, right now

His accent makes it that much better

What is a Manifold? - Mikhail Gromov

►

John Rennie

5:18 PM

I have to go chaps. A cold beer, my armchair and a copy of Transition by Iain Banks are calling to me.

Danu

"sometime again so I think that Francisco confirm indecency patient appears"

@JohnRennie Seems legit!

0celo7

@Slereah I call Einstein "compact kitty"

he is a closed, bounded subset of $\mathbb{R}^3$, so that's actually right

Secret

@slereah Pretty hazy now, I breifly learnt that in my multivaribale calculus couse in 2nd year, It has something to do with points lying within eahc other's neighbourhood???

and "closed" something?

Slereah

Well, do you know about open sets

0celo7

"compact" means heuristically "not infinite"

::waits for shitstorm::

Secret

5:21 PM

it's a set without a boundary, such as the set of points $x^2+y^2 < 1$

Slereah

basically, yes

ACuriousMind

@0celo7 It's fine, because it does indeed mean that heuristically.

Slereah

Well you can have non compact sets that are finite pretty easily!

0celo7

@ACuriousMind i am the topology master

Secret

I also heard about a clopen set, and the calculus professors often quote the real number line as an example

0celo7

5:24 PM

what school talks about clopen sets in calculus

Slereah

Trivially yes

ACuriousMind

Every space is clopen in its own topology, it's part of the defintion.

Slereah

The entire set is clopen

Secret

it is closed because everything is inside, it is open because you cannot find any boundary points

that's what my memory rmbs

Slereah

or alternatively, because the complement is the empty set :p

Which is closed

or open

0celo7

5:25 PM

the empty set is also clopen

Slereah

you can even have sets that are neither closed nor open

ACuriousMind

@Slereah How's that? You just choose one element of the cover for every point, then you have the finite cover, hence the set is compact. I don't think you meant to say that we can have a finite set that is non-compact.

Slereah

I think?

I forget

ACuriousMind

@Slereah Of course.

Half-open intervals, for one.

Slereah

ah yes

ACuriousMind

5:26 PM

$[a,b)$.

0celo7

can you have sets that are closed open and neither

Secret

I tend to remember empty set this way:

(below is not mathematical rigorous)
It is a set that whatever properties obeys the "structure" "For all x ... blahblahblah", it is vacously true

Slereah

@ACuriousMind isn't any finite open set not compact

0celo7

@Slereah yes

that's Heine-Borel?

Maybe Bolzano-Weierstrass?

ACuriousMind

@Slereah No. Compact means "Every open cover has a finite subcover". Finite set trivially have finite subcovers for every cover.

Slereah

5:27 PM

Bolstrasse-Weierzano

ACuriousMind

@0celo7 No. There are no finite open sets in $\mathbb{R}$

I think you might non be talking about "finite sets" :P

0celo7

finite set = bounded set

ACuriousMind

(finite sets are sets with a finite number of elements)

0celo7

but I see how you're using it

yes yes

see now even topology defintions make me look like an idiot

ACuriousMind

finite and bounded are not the same!!!

0celo7

5:29 PM

OK

CALM DOWN

Slereah

Well yes, but

YOU KNOW WHAT WE MEAN

ACuriousMind

And the heuristic should indeed be "not bounded" and not "not infinite"

0celo7

oh shaddup

I'm leaving

Secret

e.g. the elements in the empty set obeys associativity, this is vacously true because there are no elements

ACuriousMind

@Slereah NO I DIDNT

PROVE ME WRONG

Slereah

5:29 PM

My favorite one is that the empty set is a manifold of any dimension

Danu

@Secret Associativity is something about an operation, not about elements

Slereah

Because there is always a bijective map between the empty set and R^n

no matter the n

well, a subset of R^n

open set

Since the empty set is an open set of R^n

0celo7

@ACuriousMind are your socks mismatched

Secret

Is is true that whatever description that is written like "<some description> for all elements" it is automatically vaciously true for the empty set?

Slereah

It should be, yes

Danu

5:31 PM

@Secret Of course, because of the logical properties of implication

If $A$ is "$x$ is an element of the set", then $A\implies B$ is always true, logically speaking, when you're talking about the empty set

Secret

@Danu ah, now I see it in the more rigorous context

@Danu about associativity, ok

ACuriousMind

@Secret: The empty set is the set $\emptyset$ with the property $\forall x : x \notin \emptyset$. You would not talk about "vacuous truths" or any such things, it's simply the set that contains nothing.

@0celo7 perhaps

Secret

@ocelo7 Our uni taught set theory in year 1

the basics

0celo7

@ACuriousMind oh they're in a superposition of matched and mismatched

Slereah

My favorite definition of the empty set is $\{x \vert \forall x. x \neq x\}$

all better

Danu

5:36 PM

A conversation between Yitang Zhang and David Eisenbud

►

ACuriousMind

@Slereah That's not an allowed definition, you must say from which set the $x$ come. (This is the prescription to avoid Russell's paradox)

0celo7

damn paradoxes

Secret

Is it possible or meaningful to define a mathematical object such that no elements is equal to itself?

0celo7

@Secret there are no elements for which that is true

thus the set has no elements

Secret

ah I see

0celo7

5:39 PM

idk about Russell's paradox

Slereah

@ACuriousMind : I've seen that definition in a fair amount of math books

0celo7

no engineer has ever worried about that

Slereah

The avoidance of Russel paradox is just that condition of ZFC

I forget which

0celo7

what on Earth is this ZFC

Slereah

Zermelo Fried Chicken

0celo7

5:41 PM

brb time to find out what Lang does for a living

Danu

@0celo7 Zermelo-Fränkel axioms + axiom of Choice

Slereah

apparently it is that one :

∃y∀x(x ∈ y ↔ (x ∈ z ∧ φ))

But I forget how it works

ACuriousMind

@Slereah The axiom of restricted comprehension. It says you may not use any formulae you want to define a set, but you have to apply the formula to a set to define a subset.

Slereah

Haven't done logic in a while

ACuriousMind

It means you can't write $\{x\mid \phi(x)\}$ to define a set, you have to start with a set $X$ and write $\{x\in X \mid \phi(x)\}$ to define a set.

Slereah

5:42 PM

But then how do you define X

Secret

Axiom of choice can make many weird things, such as proving there exists a hamel basis for $\mathbb{R}^\mathbb{Z}$ but not giving any details on how to explicitly construct one

Slereah

Yeah axiom of choice is kind of the bane of constructive mathematicsd

Along with the axiom of contradiction

ACuriousMind

@Slereah You have to have some axiom that gives the existence of at least one set as a given

Slereah

Is it the... set of N in ZFC?

From what I remember

I'm a bit fuzzy on ZFC

ACuriousMind

Then you can define the empty set your way, and by the axiom of extensionality (sets containing the same are the same), this empty set is unique.

Slereah

5:48 PM

Axiom of choice is weird but then the negation of choice also gives weird results

You end up with sets with no cardinalities

ACuriousMind

I think the axiom of infinity gives usually the existence of a set, but the formulations I can find seems to presuppose the existence of the empty set already...

Secret

A video watched in the past
https://www.youtube.com/watch?v=23I5GS4JiDg
How I learnt about the long line the 2nd time, after being utterly confused by the definition in wikipedia when I 1st learnt about it,
after that video and look in wikiepda for the defintion again, the long line kinda makes sense to me

ACuriousMind

Some people also take the existence of the empty set as an axiom and build everything from there

Slereah

Yeah that is another problem

People construct set theory in various ways

Even within the realm of ZFC

ACuriousMind

6:11 PM

Why is this the 404 image for chat.stackexchange? :D

Slereah

there are much better TNG images to use

Or, the best

vignette4.wikia.nocookie.net/memoryalpha/images/d/de/…

ACuriousMind

Have you stored those somewhere or did you just search for "Q"? :D

Slereah

Q is a good search term for fun images

Also anything involving Wesley and sharp weapons

and of course the best scene of star trek

Star Trek: Voyager - The Doctor Sings for Tuvok

►

Danu

6:30 PM

@ACuriousMind There's a sentence in my diffgeo notes that doesn't make a whole lot of sense to me: Maybe it is somehow related to a German way of stating it. Can you identify the meaning?

"A derivation $X$ recognizes the vector field $X$"

Secret

Extending on Danu's explanation, Does the set $S=\{ x \in S | \text{B is always true} \}$ contains elements besides the empty set and B is some arbitrary statement?

Danu

What exactly is meant by "recognizes"?

@Secret $\emptyset$ is the only set for which the following holds: For all properties $B$; if $x\in S$ where $S$ is a set, then $x$ has property $B$

This cannot hold whenever $S$ contains an element, for otherwise the property $\neg B$ would not hold, but $\neg B$ is always a property whenever $B$ is

Secret

but then you have $\emptyset \in S$ and $S=\emptyset$, so $\{\emptyset\}=\emptyset$ ?

how to resolve this?

Danu

@Secret $\emptyset\notin\emptyset$

$\emptyset\subset\emptyset$

Slereah

actually no set contains itself

Danu

6:35 PM

but not an element of

Slereah