The h Bar

7:05 PM

lol

it's funny how annoying the particle on a ring is

hahah

> The degree of pathology exhibited by this Green's function is entertaining, especially in view of the elementary nature of the example.

indeed

" What could possibly go wrong?"

@Semiclassical exactly

I think you could also link that example up with hyperfunctions mumbo-jumbo

7:07 PM

it's a lesson in "just put a (whatever) in front of your $\sum_n e^{inx}$, you'll probably be fine"

@Semiclassical wtf is a hyperfunction

Hyperfunction

In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato in 1958, building upon earlier work by Grothendieck and others. In Japan, they are usually called the Sato's hyperfunctions. == Formulation == A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the upper half-plane and another on the lower half-plane. That is, a hyperfunction is specified...

Penrose had them in TRTR

I've never tried to understand the higher-dimensional versions because no just no

but the one complex variable version is reasonably interesting

ooooof, yeah, I guess that might work for the Jacobi thetas

nLab has a good page on them: ncatlab.org/nlab/show/hyperfunction

(with a reasonable minimum of nLab-wtf going on)

but beh-jee-sus, in the strongest Perry Cox terms possible, why in God's green Earth would you want to do that?

Jacobi thetas are really really bad because it has no analytic continuation outside a disk

7:10 PM

I remember a case where they were actually of reasonable interest

total monsters

alas, that was because there was a review article which was pointing in that direction as a place for further study

lemme find it, though

right, this article: people.maths.ox.ac.uk/trefethen/publication/PDF/2014_149.pdf

which, i should preface, is a numerical analysis review.

a good one, but that's the context

CooperCape

when they send you an email with a pdf and the links are broken... (thanks)

I've seen them used quite recently by some guys in my group

arxiv.org/abs/1708.08903

anyways, if you look at footnote 11 on page 44

7:12 PM

where thankfully the $|q|\to1$ limit is meaningless

it includes the following (somewhat obscure) remark

"These formulations can be regarded as uses of the theory of hyperfunctions, which are generalized functions defined by analytic functions in the lower and upper half-plane that may have singularities or discontinuities along the real axis. Table 14.1 is thus a list of hyperfunction approximations to the hyperfunctions of Table 14.2."

alas, that's all they say in there

I see the problem of extending certain functions beyond their domain of definition by means of beating the function up to the point of death in the hell's most demented corners a lot in physics

I guess this is a standard issue in PDE theory

@BalarkaSen fyi, the nLab page contains a sheaf-theoretic definition of hyperfunctions :P

lolol

maybe i'll check it out

what are they, to a mortal man like me?

lemme find the toy example implied by the footnote

7:16 PM

okok

okay. Let $f_N(z)=\frac{1}{2i z}\frac{1+z^N}{1-z^N}$

mhm

this has poles at $z=e^{2\pi i k/N}$, with residue...

ugh, tedious

anyways. suppose you take an integration cycle consisting of two contours, one winding just inside the unit circle and the other just outside, in opposite directions

2physics

@PrathyushPoduval yup, I think so.. I guess it should be called displacement flux, instead. thanks for your answer.

so that the area enclosed has that cycle as its boundary

7:23 PM

OK

if I then compute $\oint g(z)f_N(z)\,dz$, I'll get a weighted sum of $g(z)$ evaluated at the points $e^{2\pi i k/N}$

and this turns out to be equivalent to doing an $N$-point trapezoidal approximation of $g(e^{i\theta})$ on the interval $[0,2\pi)$

that's the motivation behind picking $f_N(z)$ that way, so that integration of it is equivalent to a trapezoidal rule.

so that's one side of this. To get to hyperfunctions, though, consider how $f_N(z)$ behaves as $N\to \infty$ for $z$ inside vs. outside the unit circle.

If we're inside the unit circle, then $z^N\to 0$ as $N\to \infty$. Hence inside the unit circle we have $f_N(z)\to \frac{1}{2i z}$

but outside we instead have $z^{-N}\to 0$ as $N\to \infty$. Hence $f_N(z)\to -\frac{1}{2i z}$

"Numerous authors have corroborated over the years through lengthy but straightforward calculations [5], [6], [7], [8], [9] that there exist an infinite class of solutions to the vacuum SSS Einstein’s equations R_μν = R = 0 for an arbitrary family of radial functions R(r) of the type displayed above ( but the curvature Riemnan tensor
R_μνρσ != 0 )."

Whaaaaat

Well I'm not that shocked

And that's the punchline: Depending on whether we're outside the unit circle, the limit $N\to \infty$ gives us one of two analytic functions @BalarkaSen

But I did not know of those

and there's an explicit jump across the boundary

for finite N, the sequence of poles serves to 'approximate' this boundary

7:30 PM

Oh man

I totally forgot

Schwarzshild wrote a paper on cosmological geometries 15 years before GR

"Schwarzschild's paper is strange to modern eyes, however, in that, when he considers positively curved space, he only discusses RP^3, which he calls \the simplest of the spaces with spherical trigonometry." In fact, he explicitly rejects S^3 as a physically acceptable model for spatial geometry, on the grounds that the light emitted from a point in S^3 would collect again at the antipode, and 'one would not consider such complicated (sic) assumptions unless it were really necessary'."

dmckee

@BernardoMeurer That is beautifully disgusting and the author is an evil genius.

@Semiclassical Ah I see

So it's a limit of holomorphic functions so that it's holomorphic on various connected domains separated by curves I guess

right

if it divides C into two regions then you define it as a pair of functions (f,g)

@Slereah whistles nice

dmckee

@Blue Well, there is a more obsfucated way to do it (or rather to find the vowels so you can not count them..

7:36 PM

with f describing its behavior on the first part of C and g its behavior on the second

@Semiclassical right

@BalarkaSen The universe is pretty obviously the projective plane

more precisely, you define it as an equivalence class with (f+h,g+h)~(f,g) where h is analytic on both of them

well, projection plane is RP^2

you mean the space

and in that sense it's defined in terms of the jump f and g on the boundary

(i don't really know why they do it like that. something to do with the sheaf formulation)

7:37 PM

@Semiclassical Oh

That makes sense

so it's like a germ of jumps in a sense

whatever that means

did you look at the sheaf definition?

They seem to use sheaf cohomology, which I don't know yet

ah, okay

@Slereah is the universe orientable

one of the reasons I like it is because it seems to really take advantage of the fact that Fourier series on a real interval = Laurent series on the unit circle

7:40 PM

@BalarkaSen yes

that is true

@Slereah How do we know that

@BalarkaSen Good hints from the CPT group in particle physics

You can't have (globally) such groups making sense if space isn't orientable

CPT is funny

Oh, tell me more

moar moar im not yet satisfied

there are a few important discrete symmetries in particle physics

Charge-conjugation, parity transformation, and time-reversal

7:42 PM

Hm what do they do

"The laws of electromagnetism (both classical and quantum) are invariant under this transformation: if each charge q were to be replaced with a charge −q, and thus the directions of the electric and magnetic fields were reversed, the dynamics would preserve the same form. "

oh i see

However, while E&M, the strong nuclear force, and gravity are all C symmetric, the weak force isn't

huh

@Slereah So how do you know you have this globally

It could be true individually on each chart

Well yes, but you know

7:45 PM

but maybe they don't patch up

Copernician principle

Parity symmetry is doing a spatial inversion through the origin

IIRC if spacetime wasn't orientable you wouldn't have spinors, you'd have two pinor fields for each particles

Which probably looks like the same thing locally, but I dunno what effect it would have

hmm

The weak force violates parity symmetry, and one can moreover find instances where CP is violated

however, in that scenario one finds that if you include a time-reversal operation, then the combined operation of CPT together is not violated

7:48 PM

@dmckee Right?!

and indeed we don't have any examples of interactions which are not symmetric under CPT

Who needs VHDL

Imagine if I had delivered

there are reasons why one would expect this, coming out of general principles in QFT

all of my Computer Architecture Assignments

in that

How glorious

dmckee

::chuckles::

Your instructor would have shaken you by the hand and then hit you with an axe.

7:49 PM

@Semiclassical curious

I think I would have effectively killed the TA

"These proofs are based on the principle of Lorentz invariance and the principle of locality in the interaction of quantum fields"

so CPT seems like a fairly well-protected symmetry

I don't know how this connects to orientability of space, though

It's pretty hard finding informations on the Einstein static universe

Almost nobody has discussed it since the 30's

@EmilioPisanty I sorta wonder what happens to that propagator when you include an interaction like $V(\theta)=V_0 \cos \theta$ (or whatever periodicity it should be)

and what happens in the limit as $V_0\to 0$

Anonymous

Is it possible to add diagrams/schematics to my Github files? I don't see any image upload option...

Anonymous

8:04 PM

I'm trying to organize the hardware schematics in one place along with the code

Anonymous

@BernardoMeurer Need halp

@Blue What's up?

Ah, yeah

git add FILE

git commit

git push

git/GitHub doesn't care about filetypes

it's all just data

Anonymous

Wait, how do I add a file with images?

Anonymous

I'm very new to Github

Use the CLI

I don't know how to do things via GitHub's website

8:06 PM

I know that there's some weeeird stuff as a periodic potential goes to zero, though

I just use git

Anonymous

@BernardoMeurer Oh, I was using the website

Anonymous

I should learn git soon

Mandatory XKCD

I considered posting that, then thought it too cliche :P

Anonymous

8:08 PM

Oh, found it....I can use "Upload Files" to upload any type of file...that should do

@EmilioPisanty also, Jacobi's theta function does show up in Graf's Intro to Hyperfunctions book

@ACuriousMind Papa bless

How have you been?

ugh, ignore that. i misunderstood what he was using it for---it's a coincidence not a sign

Apparently hyperfunctions arose very naturally in trying to solve the multi-dimensional Mittag-Leffler via the C-R equations for functions with prescribed poles, with sheaves also originating in this problem, power of complex analysis

Scariest thing I've seen is a penrose transform of these monsters

@BernardoMeurer Fine :) Doing some activities for the freshers this week

8:11 PM

@ACuriousMind Cool, scare them!

hyperfunctions in multiple variables are scary

crush them

Freshman need horror otherwise they don't develop

Yes, crush them

And don't worry, they'll develop Stockholm syndrome

@BernardoMeurer This year I'm mostly doing the fun stuff (i.e. bar tour ;) )

I sure did in Lisbon

8:12 PM

Yet people can draw pictures of them, we are weak :p

@ACuriousMind "bar tour" God I miss Europe sometimes

@ACuriousMind rigorous bar tour.

@BalarkaSen by comparison with nLab, here's the Encyclopedia of Math definition: encyclopediaofmath.org/index.php/Hyperfunction

@IcEmybReaD Yes, we will prove the existence and uniqueness of several bars.

the notation breaks my brain

8:13 PM

i have boycotted that website

why does that website look like a 1950's book turned into a website

2

lol

that's a pretty apt description

Yeah why is the latex on there looking weird

Is it even latex...

it's some weird ass latex people used in the 17th century

fucken codecogs images

@BernardoMeurer Not sure whether I want to gloat or to commiserate

8:16 PM

i dunno

png...

see, codecogs images

@ACuriousMind I have not had this little drinking ever since I was 14

Really

I have less than a beer a week

@Semiclassical that looks like random formulas for a stock image that tutoring companies use.

8:17 PM

The guy just solved a bunch of statics problems on that board, hold my beer...

nah

it's the big blackboard from "A Serious Man"

startribune.com/…

Wow Coen Bro's + physics

Howdy

if you look at the picture above closely, you'll see

maxwell's equations in integral form, the magnetic field lines of a finite solenoid

I also spy a Gaussian approximation of the delta function

wiki: 'hyperfunctions can be thought of informally as distributions of infinite order', makes some sense if you're solving the CR PDE's for solutions with poles.

8:21 PM

@BernardoMeurer "this is git. It is too complicated for its own good unless you're a brain-addled bash maniac. Use hg instead."

lol

@EmilioPisanty I'm a bash maniac :)

I know

then shine on, you crazy diamond

For everyone else, there's hg

8:22 PM

Apparently I awaken every time there is git talk

Hmm

@Semiclassical yeah, that's a good question, I wonder what'd happen

I don't feel I have enough time to add value to Physics SE

Maybe the writers just said lets solve some problems from Jackson on a board to fill it up

the trouble is that when you do the cosine potential, you get mathieu functions

and those suuuuck

So sorry for just popping up every once in a while

V v busy

8:23 PM

a more tractable version would perhaps be to do the one-gap potential

In the worst case, you'll get an irrecoverable limit, like trying to get plane waves out of Airy functions in the limit of zero potential

But you'd hope it doesn't get to that

right.

Though yeah, you're right, I'd heard of that Mathieu connection before

yeah

It might be a good project to learn those

8:25 PM

the thing is that while the cosine potential is 'easy' from the Fourier analysis POV (in the sense that it 'just' becomes a linear algebra problem on an infinite matrix)

Kiarash

Hey friends. just a non-standard question out of curiosity. is Feynman diagram related (in a fancy deep sense) to cellular homotopy?

in real space, you get Mathieu's equation, and that's got an irregular singularity at infinity

@Semiclassical yeah, you get a single - banded matrix

But that's probably one band too many

well, it's banded, but the diagonal part goes like $k^2$ where $k$ is the distance from the center of the matrix

@Semiclassical do you need to go to infinity, though?

@Semiclassical no, it's banded in the potential-free eigenbasis

8:28 PM

well, here's the relevant point from WP: "This has two regular singularities at $t=-1,1$ and one irregular singularity at infinity, which implies that in general (unlike many other special functions), the solutions of Mathieu's equation cannot be expressed in terms of hypergeometric functions."

@ACuriousMind there is a conversation I am trying to find from around spring 2015 where you gave your opinion of reading math/physics textbooks. but I can't seem to find it. do you know what I'm referring to?

@EmilioPisanty right. but if you have a cosine potential then only the first off-diagonal terms are nonvanishing, and they're all equal

so it's still banded, just not singly-banded

@Semiclassical yeah, that's what I meant by single-banded

right

As in, a single off- diagonal band

8:34 PM

?

But that still sounds like too much

in the potential-free case it'd only have entries along the diagonal

Yeah

the potential makes both of the neighboring off-diagonals nonzero

unfortunately, while that's easy enough to do numerically

it's pretty useless for actual computations of, say, level splittings

for that you have to put your semiclassical hat on and you'll end up getting exponentially small level splittings from instanton action integrals. but ugh I don't wanna remember that stuff

the search function for the chat doesn't show all instances of a word... I am sure of it.

8:37 PM

(if your cosine potential has deep wells, then you just do the usual instanton action. if your energy is very large compared to the potential, then you end up doing instantons in momentum space)

The potential is the contiguous entries

Two identical bands

right

Why would you get instantons?

in which case?

What are you tunnelling across?

8:40 PM

if it's a deep well, then the barrier would be at $\theta=0$ since the potential is $V=\cos \theta\approx 1-\theta^2/2$

so you'd be tunneling around the ring

@Semiclassical tell you what, ask on main and I'll sponsor a bounty

heh, maybe when I get a chance

for a really hilarious version of this: Take the potential to be $V=e^{2i\theta}+e^{-i\theta}$.

@Semiclassical tunnelling back to the same region you started from? Yeah, I guess that could be a thing

@EmilioPisanty It's more sensible when you think of it not as tunneling on a ring but rather that you've got a periodic potential, so an infinite number of wells

@Semiclassical start off with a non-hermitian hamiltonian and you deserve whatever it is you get

8:42 PM

lolol

obviously, that's not going to be quantum mechanics

but, it is symmetric if you do a parity transformation and then Hermitian conjugate. this has the implication that all the eigenvalues are either real or come in complex conjugate pairs

@Semiclassical no, I'm not sold that that helps

@Semiclassical oh God, please let's not do PT symmetric qm

hahaha

tbh I don't want to either.

the main reason it was nice in the problem of interest was 1) it guaranteed that the partition function you got was real

and 2) it turned out you could still do Bohr-Sommerfeld quantization in spite of the lack of hermiticity

which was ...weird

but cool

gtg bro

l8ter

Anonymous

Dayum...it worked now

Anonymous

8:47 PM

Strange

i came to the conclusion that the air quality indicator on my air purifier is rigged/ not actually doing anything... it says the air is really clean for like 5 hours then it suddenly says it is extremely dirty for like an hour and then goes back. even though nothing has changed in the room during that time.

David Z

@Kiarash no idea FWIW

@IcEmybReaD No, I can't recall what you might mean

@Kiarash What makes you think there might be a connection?

In particular, the cellular homotopy of what space do you have in mind?

@ACuriousMind you said something along the lines of "something something people who read textbooks will be able to learn stuff at a much faster rate than lectures something something and also have a deeper understanding of the subject by knowing the literature"

I believe it was in response to 0celo7.

I remember it clearly but I can't find it in the search!

I don't really care about the quote that much but I wanted to re-read that conversation because I remember remembering to want to read it for some reason and bookmarked it on my old laptop.

@ACuriousMind does that sound familiar to you / do you have any idea?

9:14 PM

@Icemybread Doesn't ring a bell, I'm afraid

@ACuriousMind what?

what's a cellular homotopy?

I managed to misread and miswrite homotopy <-> homology, I think...

@Semiclassical I'm now supposed to give 3 talks at the seminar

thankfully one is next semester

oof

@Semiclassical and my advisor wants me to do 2 more

9:18 PM

@0ßelö7 There is the cellular homotopy theorem, but I doubt that's what's meant here. @Kiarash Did you mean cellular homology?

and then 2 in my PDE class

wtf

double oof

@ACuriousMind Is that the one that says a homotopy of attaching map leads to a homotopy equivalence of results?

@0ßelö7 Nah, it says something about a map between CW-complexes always being homotopic to a cellular map

@ACuriousMind I have 22 days to write material for 2 1-hour talks

is that enough?

9:25 PM

If you already know all the material, I'd say that's enough

@ACuriousMind I don't.

Then it might get a wee bit stressful ;)

maybe

@ACuriousMind Well, I know exactly what it's about and I just gave a sloppy 1-hour talk to the organizer

he told me to give more details and stretch it to two talks

@ACuriousMind I need to mention the $\hat A$ genus but I barely understand what it is

what is it?

Beyond it showing up in the gravitational anomaly polynomial, I don't really know anything about it

@ACuriousMind What's a spin manifold of dimension $4k$ such that $\int\hat A\ne0$?

9:37 PM

@0ßelö7 Don't know that off the top of my head, either

bah

I will have to look in the Spin Geometry book

not happy

@ACuriousMind that's the cellular approximation theorem to me

that's what I was thinking

I think that's what Hatcher calls it

@BalarkaSen apparently Schoen and Yau published their proof earlier this year

aha

@BalarkaSen apparently in response to someone famous calling them out and then publishing their own proof

9:46 PM

lolol

Anonymous

What does "wavelength of antenna" mean ?

Anonymous

Does it refer to the wavelength of the radiation?

@BalarkaSen

lolololololol

this is fucking awesome

40 thousand keks to Bernardo

Anonymous

I don't get the joke

Anonymous

9:57 PM

I'm so outdated

Anonymous

What's that weird t-shirt? :P

I get complimented on this shirt every day I wear it

@Blue normies are not supposed to understand the joke

It's a |>|_|UuUuUzzi MAGNET

Anonymous

@BalarkaSen Duh, I'm not even a normie

Anonymous

9:58 PM

I don't know most things normies know

@Bernardo You look like a thicc vegan

I'm a thicc boi