The h Bar - 2018-01-24 (page 1 of 2)

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12:03 AM

@skullpatrol 1930s I guess

Jesus Christ @BalarkaSen that was a disaster

@Semiclassical yeah, after thinking about it, me too.

Not sure why I thought that tbh

I was thinking "the atomic age."

Well, that’s not so far off from 1959

12:12 AM

@EmilioPisanty wait why did you ping me? was it an accident?

:|

2 hours later…

2:10 AM

wow

(removed)

2:36 AM

Ah, hell. Ursula Le Guin died.

Now I'm sad.

ikr

R.I.P.

@Slereah My first modem was 300 baud. I could type faster than it could transmit without much trouble.

Before that my very first computing experience was on a teletype terminal my father brought home from work occasionally. It talked to the mainframe through an acoustic coupler at 150 baud.

Acoustic coupler

In telecommunications, an acoustic coupler is an interface device for coupling electrical signals by acoustical means—usually into and out of a telephone. The link is achieved through converting electric signals from the phone line to sound and reconvert sound to electric signals needed for the end terminal, such as a teletypewriter, and back, rather than through direct electrical connection. == History and applications == Prior to its breakup in 1984, Bell System's legal monopoly over telephony in the United States allowed the company to impose strict rules on how consumers could access their...

2:51 AM

Wow!

talk about the influence a father can have on his son:

> I was going to engineering school but fell in love with physics. When I told my father I wanted to be a physicist, he said, "Hell, no, you ain't going to work in a drugstore." I said, No, not a pharmacist. I said, "Like Einstein." He poked me in the chest with a piece of plumbing pipe. "You ain't going to be no engineer," he said. "You're going to be Einstein."

Suskind's father.

2 hours later…

5:02 AM

@Slereah "S. Klainerman (lecture notes in preparation)"

1 hour later…

user228700

6:23 AM

Hello, everyone :-)

Morning :-)

I think everyone is just the two of us at the moment ...

user228700

How's it going?

user228700

@JohnRennie Lol, yes.

It's stormy in the UK right now. The wind is howling round my house.

Storm Georgina

user228700

Oh, wow. So many storms!

user228700

6:29 AM

(Sorry about the delay; I was momentarily distracted by a heartwarming video I stumbled upon just now)

2

there was a tsunami watch off the Alaskan panhandle

the entire west coast of north america and hawaii were put on alert!

user228700

Oh, what? I see.

user image

@Kaumudi.H I can't believe 3,177 viewers gave that a thumbs down.

user228700

6:44 AM

Some of them believe that it is a propaganda issued by the company.

kk

could be

but still, we're talking about suicide because of an exam...

user228700

Hmm, well :-/

...I'm not saying there aren't deeper issues here

anyway, thanks for sharing the video :-)

user228700

Right, sure :-)

user image

6:56 AM

Just spent an hour coding with a Russian girl remotely. It is official ! I am stupid and have to read and code for a few more years.

user image

7:44 AM

Response to Garrett Lisi and his E8 lie group theory

I want my nobel prize for correlating particles to spaces on an adequately sized and shaped chessboard after him

8:12 AM

"the operator $f \mapsto X_p(Yp)$ defined on $C^\infty(p)$ does not define a vector at $p$"

Why not

Oh wait, is it the Leibniz rule

What does $f \to X_p(Yp)$ mean, is $Y$ a vector, where is $f$

I forgot the $f$

$$X(Y(fg)) = X(f Yg + g Yf) = X(f Yg) + X(g Yf) = f XYg + Xf Yg + Xg Yf + g XY f$$

Yeah that shows $XY$ doesn't act as a derivation, where you define a vector field as a derivation on the algebra of $C^{\infty}$ functions

Hello @JohnRennie . Please visit the problem solving room if you don't mind.

\begin{align}
(XY)(fg) &= X[(Yf)g + f(Yg)] = (XYf)g + (Yf)(Xg) + (Xf)(Yg) + f(XYg) \\
(YX)(fg) &= Y[(Xf)g + f(Xg)] = (YXf)g + (Xf)(Yg) + (Yf)(Xg) + f(YXg) \\
[X,Y](fg) &= (XY - YX)(fg) = (XYf)g + f(XYg) - (YXf)g - f(YXg) \\
&= ([X,Y]f)g + f([X,Y]g).
\end{align}

8:27 AM

Yeah I see it

Apparently you can rewrite the EH action via vielbein's as $S = \int d^D x \sqrt{-g} R = \int d^D x e R(e_{\mu} \, ^a, \omega_{\mu a} \, ^b (e)) = \int d^D x e (C_{ca,} \, ^a C^{cb,} \, _{b} - \frac{1}{2} C_{ab,c} C^{ac,b} - \frac{1}{4} C_{ab,c} C^{ab,c})$ with $C_{\mu \nu} \, ^a = \partial_{\mu} e_{\nu} \, ^a - \partial_{\nu} e_{\mu} \, ^b$, but what does $C_{ca,} \, ^a C^{cb,} \, _{b} $ mean, i.e. the , and '

@0celo7 ~~able~~ challenged

, means a partial derivative

with respect to what

the index that follows it

$T_{,a} = \partial_a T$

8:35 AM

Yeah but I don't see how that makes sense given the way $C$ is defined, if $C_{\mu \nu} \, ^a = \partial_{\mu} e_{\nu} \, ^a - \partial_{\nu} e_{\mu} \, ^b$, then $C_{\mu \nu,} \, ^a = \partial_a (\partial_{\mu} e_{\nu} \, ^a - \partial_{\nu} e_{\mu} \, ^a)$?

(The $b$ should be an $a$ in the $C_{\mu \nu} \, ^a = \partial_{\mu} e_{\nu} \, ^a - \partial_{\nu} e_{\mu} \, ^b$ above, i.e. it should be $C_{\mu \nu} \, ^a = \partial_{\mu} e_{\nu} \, ^a - \partial_{\nu} e_{\mu} \, ^a$)

Not a clue

Time to face the vile vielbein's head on now :(

Hm

I kinda want to try to define GR axiomatically

But

I'm not sure how easy it would be

I'm not sure manifolds even form a set

Or if they can only form a category

Although since nice manifolds can be embedded in $\Bbb R^n$, I think they are nice enough to form a set

Apparently second-countable manifolds up to homeomorphisms are a set

You can get both calculus and manifolds to be part of commutative algebra springer.com/gp/book/9780387955438 and interpret this insanity in terms of "observables" and "measurements" so maybe that'a route to it

'reality is a projective module'

There's a review here amazon.com/Smooth-Manifolds-Observables-Graduate-Mathematics/dp/…

Though maybe I should just learn category theory

Would probably be simpler

9:03 AM

"The process of going from sets to categories is a special case of categorification and the reverse process is a special case of decategorification."

I don't get how category theory works

What are the rules

I can never find just a list of category theory axioms

how do you prove the Yoneda lemma

if there are no rules

books.google.fr/…

Axioms are there

That's all you need, theoretically

11

Q: Serge Lang and categories

I was told that (Serge) Lang has never fallen in love with categories, to use a polite euphemism. Other people claim that, in some occasion, he has even declared his lack of interest in the subject in a somewhat harsh tone. However, I couldn't find anything (explicit) in his work in favor of thes...

>a set $\text{Mor}(X,Y)$

How can you have a set

if the category is too big

Aaaah

I agree, and you're supposed to be able to rephrase set theory in terms of categories...

'class'

yeah I suppose they use classes

Plus I guess that for my purpose, the objects are manifolds, which are sets

just that the category of manifolds isn't a set

Right now I find it ridiculous, but I am surely missing the logic

Category of manifolds

In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous and of class Cp. One is often interested only in Cp-manifolds modelled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modelled on a fixed space E is denoted Manp(E). One may also speak of the category of smooth manifolds, Man∞, or the category of...

'its objects are sets with additional structure'

9:15 AM

yeah

Well

What's the difference between a set and a class

A set is defined using ZFC

you can only have a set that you can construct using ZFC processes

Classes can be larger

for instance

there is no set of all sets

but you can build the class of all sets

Yeah, it seems absolutely ridiculous

class = set

yeah, but the alternative is

but wordplay

9:17 AM

Russell's paradox

Yeah, but it's like they just said 'lets pretend a set is not a set and call it a class'

It's not a set, tho

they follow different axioms

A set, you start with the empty set $\varnothing$

'a class is a collection of sets (or sometimes other mathematical objects) that'... "collection"

Then from this you build other sets

A class, you can define directly via properties, I think

"Outside set theory, the word "class" is sometimes used synonymously with "set"."

9:19 AM

Basically a set is, you have the empty set, and from this, you can use the power set, union and intersection to build other sets

any collection of things that can't be built using those operations isn't a set

So in ZFC a set is not actually a set, it's something restricted, so we can then define a class as another restriction on the idea of a set, because of a paradox, which is just pretending everything is fine it seems to me, idk

I'm happy to just call everything a set and not worry about the monsters of set theory

"The assumption that any property may be used to form a set, without restriction, leads to paradoxes. One common example is Russell's paradox: there is no set consisting of "all sets that do not contain themselves". Thus consistent systems of naïve set theory must include some limitations on the principles which can be used to form sets."

"ZF set theory does not formalize the notion of classes, so each formula with classes must be reduced syntactically to a formula without classes", maybe this is why I have no idea what a class is

"a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share", i.e. a set of sets that can be defined without monsters it seems like

That's because if you just define a set as $A = \{ x | p(x) \}$

For some property $p$

The system is inconsistent

The trick is that classes can contain sets

But classes can't contain classes

It's actually an old trick

Russell originally did set theory by defining a hierarchy of sets

Like you had sets

Sets that contain sets

Sets that contain sets that contain sets

etc etc

Of course, you don't have to use classes

you can just use set theory

and never worry about classes

But classes can be practical for some things

or you can also enlarge set theory

by using Grothendiek's axiom, for instance

to have more possible sets

that way you can get shit like the set of surreal numbers and whatnot

arxiv.org/pdf/1203.3911.pdf

this seems relevant

9:37 AM

I'll just pretend classes don't exist as ZF does for now I guess

Well

classes don't exist in ZFC

they're in like von Neumann–Bernays–Gödel set theory

Basically if you don't know if something forms a set or not, just pretend it's a class :p

the problem you have I think is just that you're thinking of sets as just collections

Everything except a bunch of barbers and the set of all sets forms a set :p

Yeah

what about the catalog of every books that don't mention themselves

The idea that a collection of things is not a set of things is...

Well that's what it was

9:43 AM

Sure, every empty book is in there

In the late 19th century

haha

Then problems appeared

the so called naive set theory

I went through a good chunk of that Halmos Naive book so

I have indoctrinated all this stuff

I half believe the axioms of set theory are just number theory properties in disguise at this point

IIRC if you remove the axiom of infinity, the theory is equivalent to Peano's arithmetic

The axiom of infinity is the one that says $\Bbb N$ exists

9:49 AM

Yeah, up until that axiom I think I got the sense you were defining things that would end up letting you do factorization and/or other basic NT stuff, I forget the details and thought it was half-crazy, will look into Peano arithmetic

⊢ ∃𝑦(𝑥 ∈ 𝑦 ∧ ∀𝑧(𝑧 ∈ 𝑦 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑦)))

Peano axioms are pretty simple

You have like

I'm happy to admit naive thinking leads to paradoxes, but this magical word class I just think it's a set where we pretend it's not

It's not magical :p

It is well defined

0 exists, definition of =, definition of the successor, induction

There was a time I really needed to know those axioms cold and took it really seriously constructing $\mathbb{R}$ from ZFC as much as I could to get practice at abstraction but at the end of the day who cares once you have the jist, string theory is where the action is at (hmm)

I never bother with foundations

just not my kind of stuff

9:53 AM

completion of a topological group where you set up completions in the context of uniform structures

It's all just basic real analysis where you get insanely pedantic

Pythagoras and throwing people off his boat is enough :p

$$0 \in \Bbb N$$ $$\forall x \in \Bbb N, x = x$$ $$\forall x, y \in \Bbb N, x = y \equiv y = x$$ $$\forall x, y, z \in \Bbb N, x = y \wedge y = z \equiv x = z$$ $$\forall a, b, b \in \Bbb N \wedge a = b \to a \in \Bbb N$$ $$\forall n \in \Bbb N, S(n) \in \Bbb N$$ $$\forall a, b \in \Bbb N, a = b \equiv S(a) = S(b)$$ $$\nexists n, S(n) = 0$$

@bolbteppa I always thought the fact that [X, Y] is a vector field is more intuitive when written in coordinates (the mixed partials cancel off, so you only remain with 1st order derivatives) than doing the derivations calculation

Just sayin'

Plus the axiom of induction

which is a bit more complicated

@bolbteppa yup, they worshiped the positive integers :p

Strictly positive

No zero

9:59 AM

thats what positive means you sad bum

anyone who uses positive to mean >= 0 is a sad bum

:shots fired:

The best thing ever is defining relations and function in terms of ordered pairs and subsets

But then that is set theory

and you obviously don't know what a set is

Enderton is a great book

What I went through of it

In the words of Gromov, a function doesn't mean anything and is undefinable

anyone who convinces you otherwise is trying to show off

(end quote)

10:01 AM

lol

i love his quotes. they're kind of idiosyncratic but not contentless

the notion of function is pretty built in on the human mind... associating stuff to stuff

(If we're defining functions as subsets of power sets then because sets are undefinable building blocks I guess that's legit :p)

yeah

come on, even a high schooler understands what a function is :P

i mean nobody really defines a set

you just take it to be a thing and impose some axioms on it

10:02 AM

Plz, sets are defined :p

The undefined notion is "collection"

it's like the notion of points on Euclidean geometry. nobody knows what a point is

Collection is the bullshit word for everything

Also "container" in computer science

@BalarkaSen Euclid defined it!

yeah but his definition was bullshit because you dont know what dimension is

Definition: that which has no parts

it was not really a definition

its an idea

10:04 AM

Plz don't badmouth the father of mathematics

^

a breadthless length

"A line is breadthless length."

Solid definition

'that which never ends' :p

length, breadth etc are not defined notions. they are ideas

10:05 AM

number is an idea

It's actually a cool style of doing mathematics. Don't define something, take it to be a thing, and impose some axioms on it

It's the so called "synthetic mathematics"

'a compactified 10 dimensional string'

The first wave of concrete mathematics was by Descartes of course

yes

Well that's how all math is done, really

in the end the primitive notions aren't defined

10:06 AM

@Slereah Yes, but not explicitly. Euclidean geometry was pretty explicitly synthetic.

you have to start some where

Qiaochu Yuan once explained to me that the idea of type theory was kinda synthetic

Types are not defined objects

Plus Euclidian geometry isn't even complete so obviously Euclid was a big fraud

Nobody knows what a type is

But you can make a theory out of it by imposing certain axioms on it

Types are easy

10:08 AM

you can ask why, why,... forever

I just googled it

user image

^there's a type

lol

what a shitty laptop

user image

Many types

nsfw

10:10 AM

nothing shameful about the human form

blood type :P

@Slereah I dislike the human form.

you're not my type

@BalarkaSen I would too if I had yours

2

10:40 AM

humanity is so overrated

why?

Captain Bohemian

I can't understand what is coalgebra through Wikipedia.

it's the dual of an algebra

An algebra being a cocoalgebra, of course

because they like to destroy themsevles

Captain Bohemian

but I don't know the dual here means. Does it means the same as dual of a Hilbert space?

10:46 AM

it is similar, yes

Captain Bohemian

so the relation between algebra and its coalgebra is like the relation between vector and its covector?

More general

It's part of the category theory definition of a dual

Any function from the algebra $A$, of the form $f : A \to X$, becomes $f^* : X \to A^*$

Or something to that effect

11:07 AM

1

Q: Noether's theorem - Making a global symmetry local (via the SEM tensor)

We are doing Lorentz-invariant Lagrangian field theory in Minkowski spacetime, and I'm now considering only the form of Noether's theorem where the fields are varied. Assume that $\delta \phi$ is a symmetry of the Lagrangian. The variation is of the form $\phi_\epsilon(x)=\phi(x)+\epsilon a\delt...

lagrangian-formalism symmetry stress-energy-momentum-tensor noethers-theorem

Too broad?

11:52 AM

Is category theory one of those thing where the theory by itself has very barebone structure

but you define specific category theories

with additional axioms

12:07 PM

Not really, my sense is that you're trying to set up concepts in a way that applies to multiple areas at once, e.g. group extensions applying to Lie algebras with modifications but the same idea 'really', or prove, say, 'Cayley's Theorem' from group theory in a way that applies to multiple structures at the same time, resulting in Yoneda's Lemma

maths.ed.ac.uk/~aar/surgery/wall.pdf

oh no

typewriter and pen

1961?!?

12:27 PM

That papers talks about some tubular neighbourhoods $\phi_i : \partial M_i \times I \to M_i$ defining a map $\phi : \partial M_1 \times D^1 \to N$

I'm guessing $I$ is $[0,1)$ since I think that's a collared neighbourhood

But what is $D^1$

From context I think that's $(-1, 1)$

Which I guess is the 1 dimensional disk

nvm

nvm, I see you already dropped the axe :P

I think they still used slide rulers in those days @Slereah

The 60's? Eeeeh I dunno

By the 60's calculators were starting to be big

Although I guess some still did

the poor scientists

:-D

LGR Tech Tales - The Pocket Calculator Wars

have some calculator history

You NERD

12:47 PM

Bringing 1916 to 2016: Slide Rules

"co-ordinate neighbourhood"

Just after Einstein's general relativity paper

1916

12:59 PM

I have found the proof that the gluing of two manifolds by their boundary is a manifold

hurray

Now to find how to do it for just one manifold!

user image

is that a sombrero viewed from inside

nah, I'm trying to say don't fall down the "rabbit hole" with that old paper :P

Captain Bohemian

I don't quite know categroy theory. Probably that's the reason why I can't understand the Wikipedia article coalgebra.

coalgebra is an anagram for a cobra leg

that is why it makes no sense

1:09 PM

@Slereah Fucking roasted

I hate to star that

But here you go

it's a classic schoolyard burn

@BalarkaSen my advisor gave me a book on harmonic functions

You should read with me

I need to read n books

where n is arbitrarily large

whats the book

I'm guessing the proof for gluing a manifold to itself will be mostly the same

@Slereah $\Bbb H^n \cup_{\partial \Bbb H^n} \Bbb H^n = \Bbb R^n$, that's all you need

1:13 PM

glue the two collars of the boundary together, get the atlas of those collars, add them to the atlases of $M$

yeah that's basically the proof

You construct a manifold that overlaps the gluing

mhm

Which is $\partial M_1 \times [0, 1) \times \partial M_2 \times (-1, 0] \to \partial M_1 \times D^1$

I still have to prove that the charts of $\partial M_1 \times D^1$ and $M$ overlap in the correct way

@Secret but there are no living creatures that don't fight amongst themselves for survival...

...as proved by Darwin.

though I think for a single manifold I should specify that the collars don't overlap, too

罗百吉 - 机车女孩 (Tai Mai Shu - Ugly Girl)

Classic chinese meme song

1:29 PM

@BalarkaSen Intro to potential theory by Helms

We're trying to understand the construction of Green functions on complete manifolds

it's mostly a mystery but this book does it carefully for R^n which might shed some light on the situation

I see

Also need to do some proofing for the other type of gluing where you glue open sets together

I'm running out of patience in the math room

because for some reason that is also called gluing and Hajicek won't shut up about it

@BalarkaSen are they dissing Euclid

@0celo7 have you ever seen
\begin{align}
S &= \int d^D x \sqrt{-g} R = \int d^D x e R(e_{\mu} \, ^a, \omega_{\mu a} \, ^b (e)) \\
&= \int d^D x e (C_{ca,} \, ^a C^{cb,} \, _{b} - \frac{1}{2} C_{ab,c} C^{ac,b} - \frac{1}{4} C_{ab,c} C^{ab,c})
\end{align}
with $C_{\mu \nu} \, ^a = \partial_{\mu} e_{\nu} \, ^a - \partial_{\nu} e_{\mu} \, ^a$

1:40 PM

@BalarkaSen again?

@Akiva This place is too toxic for you

Akiva Weinberger

Hm

Can't find it anywhere, never seen this before, nothing like it anywhere...

Like a moth to a flame...burned by the fire...

@BalarkaSen oh please

implying the math chat is any better

@bolbteppa don't think so

1:43 PM

don't dis the irrational room

Ricci scalar in terms of vielbein's in brief notation that doesn't seem to appear anywhere

Rarely if ever expressible as a ratio of integers.

I like how h bar and maths are merging. It goes in according to The Plan

By 2019, The Plan is expected to be completed, and the impossible will soon happen

Time travel?

:P

the hell is a "diffeotopy"

1:55 PM

Isotopy through diffeomorphisms

2

Q: Diffeotopy and connectedness on manifolds

Let $M, N$ manifolds without bundary and $f: M\to N$ and $g: M\to N$ embeddings. We say that a differentiable map $h: [0,1]\times M\to N$ is an isotopy between $f$ and $g$ if each of the maps $h_t:M\to N$ with $h_t(x) = h(t,x)$ is an embedding and $h_0=f, h_1=g$. By a diffeotopy of a manifold...

differential-geometry manifolds differential-topology

2:07 PM

"Finally we remark that it is sometimes desirable to glue together two parts of the boundary of the same manifold. If the parts are disjoint, ,we can use disjoint tubular nbds to effect this. "

This paper's pretty good

too bad it was written on toilet paper

So I'm pretty convinced that the cut and paste wormhole bullshit makes sense now

Though I don't know if the Israel junction thing does make sense

Also I need to check that the smooth structure is fine too because IIRC you have to smooth out the gluing

Gluing is one of those crazy things topologists do

I wonder why those tags differential-geometry etc. are so dark? @ACuriousMind

there's a lot of gluing shenanigans with weird spacetimes

2:31 PM

I wanna show that given some metric on the spacetime, if you perform some cut and paste, you indeed get the correct junction

Synge was basically doing all the GR when pretty much nobody else was doing GR

Good going Synge

How many people were doing important GR work in the 50's

Hi all

Where is it recommended that I post a question about python coding with application to quantum measurements. Is it okay to post in PSE?

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