The h Bar

12:17 AM

@EmilioPisanty Truth be told...

user54412

3:49 AM

@DanielSank So I'll be SB from noon this Sunday to Tuesday afternoon. I don't have any plans for Sunday so if you want to meet an internet stranger...

0celo7

@ChrisWhite !

@ChrisWhite say I have $f:M\to N$ and $\phi:\mathbb{R}\to M$

and I have a chart $(h,U)$

user54412

ok...

0celo7

and then I write $f(\phi(t))=(f\circ h^{-1})\circ (h\circ \phi(t))$

can I Taylor expand this like I would anything on $\mathbb{R}^n$?

because $f\circ h^{-1}$ is just a function on Rn

and $h\circ\phi(t)$ is a curve in Rn

user54412

i think so

Ghost

hello!

0celo7

4:04 AM

God save us

user54412

?

0celo7

satan is here

Ghost

ummm what?

0celo7

THE BANKER | Damien Slash

►

user54412

good to see things make as little sense around here as usual

2

0celo7

4:09 AM

does no one else see the pentagram

user54412

aren't pentagrams upside down?

Ghost

it's a pentacle ancient-symbols.com/pagan_symbols.html - a Pagan symbol, not satanic sysmbol

0celo7

Pagan = satan

Ghost

incorrect

Paganism is not satan worshipping at all

0celo7

4:25 AM

A&R GUY p3 | Damien Slash

►

@ChrisWhite How do I show the pushforward is linear?

using Arnold's definition

I think I need to show that given two vectors there is an equivalence class of curves that gives a linear combination of these vectors

hmm, so I need to give the set of equivalence classes of curves a vector space structure

Ghost

but given that I am an Atheist, the image is a series of geometric forms to me

user54412

@0celo7 what's arnold's def?

Slereah

4:40 AM

Hail Satan!

0celo7

@ChrisWhite if $\phi$ is in the equivalence class defining the vector $v$ then $$f_*v=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_0f(\phi(t))$$

@Slereah exactly!

Ghost

grow up

Secret

Random 2
https://en.wikipedia.org/wiki/Pando_(tree)

user54412

@0celo7 sounds right -- something like addition being concatenation

user54412

I'm assuming equivalence classes are just reparameterizations with the same images

user54412

4:45 AM

Also, I'm only 10% paying attention, and about 5% coherent

0celo7

@ChrisWhite concatenation? no

that doesn't make sense

I think

Secret

@Ghost Yes, paganism and satanism are miles apart

Ghost

@Secret except to the ignorant

Secret

though I still know very little about satanism other than it is often used in heavy metal.
To date I have still refused to read the wikipedia article on satanism for fear of getting my future screwed due to being exposed to some dangerous concepts

user54412

@0celo7 Mmm perhaps you're right. But now I don't really know what's going on. What are curves? Maps from R to a manifold? What equivalence relation do we have then?

0celo7

4:48 AM

@ChrisWhite my coherence and attention match yours

how about we don't do this at 1 in the morning :)

user54412

agreed

user54412

especially after I was coerced into rewatching 2001 a few hours ago

user54412

that LSD feeling lingers on

Ghost

@Secret I am an Atheist - and what I see is a pentagon, 5 isosceles triangles and 5 wedges - in a geometric pattern that is called a pentacle (or a pentagram to the ignorant) and is a symbol of Paganism (which is not satanism)

0celo7

looks like 666 to me

user54412

4:51 AM

@Ghost I count 10 isosceles triangles ;)

Secret

I am agnostic, thus I see both the religious meaning and the straightforward geometric meaning

Ghost

@ChrisWhite even better!

0celo7

I see 666 triangles :o

Ghost

@0celo7 whatever, kid

0celo7

why are you calling me kid?

Secret

4:52 AM

My current belief system is:
when in doubt, listen to what all worldviews said, and the truth is usually the weighted average of the intersections plus some outliers

Ghost

@0celo7 why are you calling me 'satan'?

0celo7

@Ghost your Halloween costume suggests it

Ghost

@0celo7 whatever amuses you

Secret

I was hardcore scientism back in 2013, and then become agnostic afterwards

0celo7

scientism?

Secret

4:56 AM

Scientism

Scientism is belief in the universal applicability of the scientific method and approach, and the view that empirical science constitutes the most "authoritative" worldview or the most valuable part of human learning - to the exclusion of other viewpoints. Accordingly, philosopher Tom Sorell provides this definition of scientism: "Scientism is a matter of putting too high a value on natural science in comparison with other branches of learning or culture." It has been defined as "the view that the characteristic inductive methods of the natural sciences are the only source of genuine factua...

basically someone who is way too into the scientific method and think it solves everything

DanielSank

@ChrisWhite Hells yes!

Wanna email me?

You can find my email via my Physics.SE profile.

user54412

sure

Secret

It's not a healthy attitude because it leaves no room for arts, culture , philosophy etc.

Ghost

@ChrisWhite that makes my profile much simpler to write

Slereah

Apparently @Ghost is rather sore at being thought a pawn of Satan

Ghost

4:59 AM

@Slereah not really

Slereah

Poor choice of avatar for this

0celo7

the silver lettering has worn off of my Visa

now it's ghetto

@Slereah he's trying to attract attention

Ghost

@Slereah not really, depens how you perceive it

Slereah

You should have taken the opposite of a satanist avatar

Like a kitten

Ghost

it is not a satanist avatar

Slereah

5:01 AM

Sure, why not

The cross is also just a geometric shape

Ghost

it's a pentagon, surrounded by 10 isosceles triangles, surrounding by a circle

Secret

Being an agnostic helps me to open to all possible worldviews.
This is how I learn a bit about Buddhism, Paganism, Hinduism, Sikhism, Taoism, Muslim etc.

However because of my inherently straw nihilistic dark personality, hence easy to get addicted to dangerous concepts, I don't think I am ready to learn about satanism yet, for fear of destroying my future

Slereah

Satanism is mostly a religion of people who like to dress in silly costumes

Ghost

I am an Atheist, I have no religion

Slereah

Secret

5:07 AM

Within the scientific community, it gives me an asset and liability to consider alternate ideas seriously, which is why I often need help form other professionals and nature itself to tell me whether I am on the right track

Slereah

Happy halloweeeeen

Ghost

to me, God, satan, heaven, hell etc etc do not exist

Secret

while to me, all of these theological entities cannot be detected by science, but we are not sure if they are compeltely ruled out of existence

Ghost

each to their own beliefs, observations, whatever

Secret

So in short, I am agnostic with a science core

which is why people often confused I am atheist

Ghost

5:09 AM

@Slereah to me, it is

0celo7

@Slereah we need to get together sometime

you coming to this side of the pond any time soon?

Slereah

Not in the near future

Secret

O btw I have two physics questions

Ghost

I am all about Science and Mathematics - so my avatar is a series of geometric shapes that I like - if others have a problem with it, too bad

Slereah

No problem

Just a tad amusing to see you so defensive

Ghost

5:14 AM

I wasn't the one that started with the childish 'satan' crap

Secret

Q1 Is there a maxwell boltzmann like distribution that relates rotational states J with populations in the rotational state?

Q2 I often heard that entropy in statistical mechanics is the no. of microstates that give the same macrostate. How is this different form the no. of degeneracy?

Q1 i.e. Given rotational states J, if there exist f such that $$\frac{N_J}{N}=exp(f(J))$$ ?

Slereah

Degeneracy isn't for macrostates

Degeneracy is the number of states for a given eigenvalue of the Hamiltonian

user54412

you're all seeming particularly... weird tonight... in an adorkable sort of way

Slereah

Maybe because it is HALLO
OWEEN

Ghost

@ChrisWhite lol 'adorkable' - love it

Slereah

5:17 AM

All sorts of spooks and goblins tonight!

user54412

I'm about to fall asleep, and now I'll probably dream about the personas in this chat room.

Ghost

I think I'll be keeping this avatar - love the symmetry and how the shapes come together

Slereah

It only has $Z_5$ symmetry

Pretty weak

Well also $Z_2$

Secret

So is degeneracy in some sense a "subset" of microstates, because of how you can have some configuration of some quantities that corresponds to some macrostates described not (or not just) by the same energy, but also other thermodynamic properties like pressure, volume?

user54412

$Z_5$ rotations $\rtimes$ $Z_2$ reflections = $D_{10}$?

Slereah

5:24 AM

Wot

Well degeneracy will contribute somewhat to the number of microstates for some macrostates

Secret

i.e. Let's assume we have some system $S$ and it has the following degrees of freedom $(a, b, c, d \cdots)$. Therefore each state $i$ in this system is described by the ntuples $k_i=(a_i, b_i, c_i, d_i \cdots)$

So for degeneracy in i it will be the number of $k_i$ such that

$$\hat{H}|k_i\rangle=E_i| k_i\rangle$$ (1)?

But since a macrostate $Z_j$ can be described by some m tuples $(A,B,C,\cdots)$ e.g. P, V, T

So we can have some microstates $k_i$ that satisfy

$$\hat{A}|k_i\rangle=A_i|k_i\rangle$$ etc.

Thus showing that $degeneracy \subseteq microstates$ ?

typo: The 2nd set of $k_i$ s after (1) should be $k_j$

DanielSank

5:50 AM

@ChrisWhite Never heard "adorkable" before.

user116211

7:15 AM

I wish each & every member of the community a great haunted Halloween:P

2

user116211

Slereah

7:56 AM

Get out the Fadeev Popov ghosts and the skeleton diagrams!

Danu

8:31 AM

@DanielSank "Advanced level question plzzzz help"

best title

user116211

9:06 AM

@Danu: Really advanced!! How could he get such creativity??

user116211

I was thinking many things when I was about to click that question & then when I noticed it, to me, it was the greatest joke of the century! Really advanced:P

user685252

:D

0celo7

2:42 PM

Hello?

Secret

probably everyone is too busy celebrating halloween

hence the silence

0celo7

What time is it where you are

Secret

1:49, so australia probably has passed halloween

0celo7

Eastern coast?

Secret

yup, sydney

Qmechanic

2:55 PM

@KyleKanos and other reviewers: Concerning physics.stackexchange.com/q/215612/2451, I stopped the migration to Math.SE in accordance with the Don't-migrate-crap-policy.

0celo7

@Secret I know someone near Brisbane so I know the time delay to the Aussie East Coast :)

AngusTheMan

3:27 PM

Qualitatively speaking can anyone explain the main differences between Dirac-Bergmann vs Jackiw-Fadeev approach to constrained dynamics?

AngusTheMan

4:12 PM

On an unrelated note does anyone have familiarity with Markov chain Monte Carlo? I am running a code I have written for global optimisation and part of the procedure involves MCMC steps. I set limits for the acceptable range that the variables are allowed to step within, I know roughly what are appropriate and I am just refining the values.

HOWEVER

...

Whenever I run this, all my variables tend to the maximum limit of their value. So I, initialise them, update them via Metropolis Hastings and they all tend to the maximum permissible value!!

Any ideas?

ACuriousMind

4:27 PM

@AngusTheMan Jackiw-Faddeev presupposes a first-order Lagrangian and is mainly interested in obtaining the correct quantized theory. Dirac-Bergmann is both an approach to quantization and the classical theory, and works with essentially arbitrary Lagrangians as starting point.

No idea for your Monte-Carlo thingy though, I suck at those.

AngusTheMan

@ACuriousMind Ah so only for first order Lagrangians then! Cheers that clears it up :) haha yeah me too actually come to think of it!

alarge

@AngusTheMan Difficult to say without seeing the code. What do you do at the boundaries anyway? Just reject the move?

ACuriousMind

@AngusTheMan Well, the "only" is not as restrictive as it sound, given a Hamiltonian $H(p,q)$ you can always make a first-order Lagrangian by Legrendre transforming, but keeping the canonical momentum as a configuration space variable, i.e. get a Lagrangian $L(q,p,\dot{q},\dot{p}) = p\dot{q} - H(p,q)$.

AngusTheMan

@alarge So its for replica exchange monte carlo, for structure refinement from diffraction data. We always have a rough idea of what the values of the crystal parameters (unit cell, microstrains, thermal ellipsoids etc = variables) to start with, so we need to keep them within a given physical limit. If I didn't put a limit on then the Monte carlo would take them way out of acceptable values leading to inefficient computation. I reject the update if it is outside them limits.

alarge

@AngusTheMan Well could it be that the values are somehow unfortunate and you are stuck in a local minimum of some kind?

AngusTheMan

4:42 PM

@alarge I could be, I mean I let it run for about 2 hours and I got several exchanges, and in general it was getting better. Maybe if I optimised it more then with more exchanges it would converge. Because it is the exchanges that allows it to get out of the local minima. \

@ACuriousMind So this is the phase space Lagrangian. Are we not inverting the conjugate momenta there?

ACuriousMind

@AngusTheMan Nope, the whole point is not doing that to keep the first order form. (First order means that the time derivatives only appear linearly, in case you thought something else)

AngusTheMan

@ACuriousMind So we generate mixed $p\dot q$ terms in these Lagrangians then? Just out of interest, surely such an object can't live in $T^*M$ for configuration manifold $M$? Maybe $T(T^*M)$?

ACuriousMind

@AngusTheMan Exactly. The phase space $T^*M$ is now the configuration space and the Lagrangian lives on its tangent bundle

AngusTheMan

@ACuriousMind Ahhhh... So if I had an $n$th order Lagrangian can I always create such first order one? Is this to do with higher order field dependence .. Ostrogradsky? etc..?

By taking successive tangent bundles to phase space - 'configuration manifolds'

ACuriousMind

@AngusTheMan You don't have to iterate - doing this once suffices: Take the Lagrangian $L(q,\dot{q})$. Legendre transfrom it, if the transform is singular add the constraints with Lagrange multipliers $\lambda$ to obtain the Hamiltonian $H(q,p,\lambda)$, now write $L(q,p,\lambda,\dot{q},\dot{p},\dot{\lambda}) = p\dot{q}-H(q,p,\lambda)$ which is obviously first-order.

0celo7

4:55 PM

What on Earth are you guys talking about?

AngusTheMan

@ACuriousMind But what about higher order field dependence? $L=(q,\dot q,\ddot q,\dots, q^{(n)})$

ACuriousMind

@AngusTheMan Oh, who does that? :P

AngusTheMan

not field... discrete coordinates

@ACuriousMind Einstein-hilbert?

;)

(I think)

ACuriousMind

If you have dependence of the Lagrangian on higher derivatives, none of the usual procedures work. I'm not sure if one can obtain a Hamiltonian picture for those

AngusTheMan

Oh yeah, the whole instability thing.

ACuriousMind

4:57 PM

And you need a Hamiltonian picture for canonical quantization. Not sure if anyone has been crazy enough to look what the path integral does with those Lagrangians

Albeit using the path integral without a Hamiltonian picutre is also ill-defined because it is usually derived starting with $U(t) = \mathrm{e}^{\mathrm{i}Ht}$

0celo7

srs what is this

AngusTheMan

hmm true! I think they are generally not physical systems, purely academic? But would be nice to develop a generalised approach which reduces to the usual for only second order. Cheers for the conversation Curious mind, you cleared a lot up in my mind :) normally I don\t get to talk about mechanics with people ...my friends being chemists and all ... ;)

ACuriousMind

@0celo7 How to generate the first-order Lagrangian for the Faddeev-Jackiw method for constrained quantization, not sure what's unclear about that ;P

AngusTheMan

@0celo7 I was asking how to reduce the order of a Lagrangian to $(n-1)$.

0celo7

@ACuriousMind what does one need this for (and I've never heard about that, so)

AngusTheMan

5:05 PM

taking classical systems ---> Quantum description

0celo7

yes, that's what quantization is

is this for gauge theory?

ACuriousMind

Yeah, the main use case of quantization of constrained systems is for gauge theories.

Some people might even use the terms "constrained system" and "gauge theory" interchangably

0celo7

so is this in QoGS

ACuriousMind

They do not talk about the Faddeev-Jackiw method, no

0celo7

well never gonna learn about it then

QoGS is on my list

but at this point physics is being pushed out of the way

@ACuriousMind so, there's an exercise in Arnold to prove the pushforward is linear. I can do this using the "western" treatment of vectors, but it's not clear at all in the equivalence class of vectors picture

ACuriousMind

5:15 PM

The "equivalence class of vectors" approach is not distinctly Russian, you can find it all over "Western" diffgeo, too. Sometimes it's convenient to think about derivations, sometimes about curves.

0celo7

*equivalence class of curves

Huy

wat

0celo7

what is your confusion, my son?

Huy

why is plex still not approved on apple tv 4 app store

i am angry

0celo7

@ACuriousMind Right, so what I first want to do is show that the "equivalence class of curves" tangent space is a vector space.

So there has to be an isomorphism $\alpha:T_pM\to \mathbb{R}^n$

Huy

5:21 PM

is trivial m8

0celo7

@Huy for you

Huy

4u2

0celo7

So let $[\gamma]$ be the equivalence class defining the vector $v$.

So $\alpha(\gamma)=v$ where $\gamma\in[\gamma]$. Injectivity: Suppose $\phi\notin[\gamma]$ but $\alpha(\phi)=v$. But by definition, $\phi\in [\gamma]$. Contradiction, so $\alpha$ is injective. Surjectivity: For each vector $v$ there must be an equivalence class. To prove this, simply construct the equivalence class, starting with one vector $\gamma^i(t)=\gamma^i(0)+v^it+O(t^2)$ and all vectors in the class differ from this one by second order terms.

@Huy Good enough?

so, given this, we can define a vector space structure on the space of equivalence class of curves

@ACuriousMind Perhaps: Let $\gamma_1,\gamma_2$ be curves in equivalence classes defining vectors $v,w$. Above we defined how $av+bw$ can be represented by another curve $\gamma$. Then $$f_*(av+bw)=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_0f(\gamma(t))=\cdots=Df‌\cdot (av+bw)=\cdots=af_*v+bf_*w$$

@ACuriousMind Something like that?

there's some chart stuff missing + other details

ACuriousMind

5:42 PM

Since I don't know how you defined half of the things there, I can't say much more than "You seem to have the right idea".

0celo7

what do you need defined?

@Huy plex?

ACuriousMind

@0celo7 Not really in the mood for nitpicking proofs, so nothing

0celo7

@ACuriousMind is there a mood for them?

Secret

https://en.wikipedia.org/wiki/Consistent_histories
>However, Griffiths[4] holds the opinion that asking the question of which set of histories will "actually occur" is a misinterpretation of the theory; histories are a tool for description of reality, not separate alternate realities.

Huy

sory 0celo7 I'm trying to make some dinner with what's left in my kitchen since I couldn't be arsed to go groceryshopping today

Secret

5:53 PM

I must be misunderstood or soemthing, but after looking at the maths of consistent histories and then recalling the very brief introduction on the concept of path integrals and its relationship with action, it still give me an impression it is some kind of alternate reality

in particular, it seems the experimental outcome that we observed is the only set of histories that are consistent

out of all the possible histories, with some probability

dmckee

6:06 PM

I'm always a little surprise to find out what basic questions we don't have a good answers for.

I'm offering a 250 point bounty on

3

6

Q: How do electrons "know" to share their voltage between two resistors?

My physics teacher explained the difference between voltage and current using sandwiches. Each person gets a bag full of sandwiches when they pass through the battery. Current = the number of people passing through a particular point per unit time. Voltage = the (change in) number of sandwiches...

Get in while the getting is good.

ACuriousMind

...sandwiches? oO

What kind of analogy is that? What's wrong with the old "It's like water in a pipe"?

0celo7

What's wrong with the old "virtual photons are whizzing back and forth"?

Secret

Random 4
https://www.google.com.au/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=multivalued+energy+distribution+in+time

HDE 226868

6:48 PM

Is anyone here familiar with bundle methods of descent, as mentioned here? It links to a page on subgradient descent, but that's obviously not a descent method.

Danu

7:24 PM

Hefty bounty you put out there @dmckee

@dmckee I feel like @DanielSank would be into this? Not sure why... probably because something something circuits something something engineering ;)

0celo7

I hear the circuits class is the hardest class in undergrad here

Danu

@ChrisWhite Might've been the LSD you were taking? :)

@0celo7 Because nobody manages to stay awake?

0celo7

@Danu That's not entirely wrong :)

truth be told I'm having a hard time staying awake doing (inverse) Laplace transforms right now

who wants to inverse transform $$\frac{s+5}{s^2-4s+4}$$

partial fractions is too hard :(

Huy

sory

my students do partial fractions too

they were really good at it

better than integrating by parts

0celo7

they hate you for it

Huy

7:31 PM

nah they love me

im the coolest teacher eva

0celo7

no

did you take a look at my solution above?

Huy

nah I'm looking at a review of the Toyota Aygo now

I need to figure out which car to buy

0celo7

bish

check it out, the proof is so elegant

Huy

lol

good one

0celo7

ok I'd like to see you prove it more elegantly using the shitty definitions I was given

Huy

7:34 PM

I'm busy watching car reviews

that's a bit more important to me atm

sory bro

0celo7

so you insult me and then leave

Huy

that's what she did

0celo7

I see how it is ::sniff::

Huy

send me 10k to spend on a car and I'll look at ur proof

0celo7

bank account?

I should send you like 1 cent :P

Huy

7:35 PM

i will report u

0celo7

what?

why

Huy

for troling

u trol

0celo7

you're crazy, you know that?

insane even

more so than the resident ayy lmao even

Huy

wat

I just want a car m8

why is that crazy

0celo7

hey get your students to partial fractions $[(s+2)(s-2)]^{-1}$ for me

Huy

7:37 PM

lol

they can do that in less than 1 minute

0celo7

teach me

Huy

ansatz

A/(s+2) + B/(s-2)

0celo7

oh, no shit

Huy

ikr

0celo7

I thought you had some super secret method

Huy

7:38 PM

no

that takes 1 minute

0celo7

I've been on it for 5

Huy

wtf

you get

$\mp \frac{1}{4}$

enjoy

0celo7

mind = blown

Huy

troll

0celo7

why do you call me this hurtful word ;_;

Huy

7:40 PM

the truth is quite hurtful sometimes

0celo7

great, I must have done something wrong...that's not the solution

Huy

because you forgot the numerator

0celo7

I didn't.

The solution of the ODE is wrong

Huy

lol

0celo7

Wrong form, even.

Huy

7:41 PM

nub

I'm looking forward to teaching complex numbers next year

then they will enjoy partial fractions a lot more

0celo7

what's a complex number

Huy

one too complex for you to understand

0celo7

ouch

Huy

involves vectors and equivalence classes

very abstract

but in the end it's just isomorphic to $R^n$

0celo7

what!

Huy

7:43 PM

ikr

0celo7

complex numbers are isomorphic to $\mathbb{R}^n$??

for any $n$?

Huy

yes

even $n = \infty$

that's the so called Riemann sphere

0celo7

hory shite

tell me more

Huy

it's a compact space

0celo7

is that the thing from string theory

Huy

7:44 PM

because whenever $n = \infty$, the space is compact

and thus all norms are equivalent

0celo7

dude how much of this is BS

XD

in any case

wat case

the case of $y''+4y'+4y=0$ and a stupid student

$y(0)=y'(0)=1$

Huy

7:47 PM

wat

why did you need that fraction before

don't you just get $\lambda = -2$

0celo7

uh, Laplace transform

because

I'm not that smart

Huy

ok

0celo7

see I get an exponential in the answer

Huy

ok

0celo7

but I can't get the $x\mathrm{e}^{-2x}$ part

Huy

7:49 PM

sory

0celo7

I can solve it by eye using the characteristic equation method

Huy

but ?

0celo7

but the directions of the problem are to use Laplace transforms...

Huy

then use it

0celo7

aha! sign error!

Huy

7:50 PM

no

an odd number of sign errors

0celo7

"a" sign error implies one, one is odd

Huy

I learned that joke from MO

very funny site indeed

don't ruin my jokes

0celo7

aha! there's the three! omg no partial fractions! woooooo

Huy

would have been too hard for the other students

right ?

0celo7

partial fractions is just about the most challenging thing there is math

Huy

7:53 PM

can you prove uniqueness and existence of it?

0celo7

god no

Huy

I had to

in first semester

I was horrified

0celo7

I have the mathematical maturity of a middle schooler, how many times do I have to say it

Huy

I'm looking at my old analysis exercises now

on the same exercise sheet there's Weierstrass function

I don't think I learned a lot from that proof

0celo7

m-hikari.com/imf/imf-2012/29-32-2012/cookIMF29-32-2012.pdf

holy fuck

Huy

7:55 PM

hue

0celo7

I could never do that

Huy

@0celo7: http://i.imgur.com/LvlIRga.png
recognize who the prof is

0celo7

no

Huy

trol

0celo7

sure, just call me that

Huy

7:57 PM

the other day u asked about a book for ergodic theory

0celo7

yeah

Huy

I sent u a book

it's the same guy

ur bad with names

0celo7

I think you're the troll

Transcript for