The h Bar

@ACuriousMind So the latter comprises CMB and more?

@0celo7 no idea what you mean

@SirCumference Yes.

@ACuriousMind So what else would it comprise?

@ACuriousMind How would you construct the Lebesgue integral?

Ok, for compactly supported functions its a limit of Riemann integrals of elementary functions in a cube, right?

@ACuriousMind

@SirCumference Uh, pretty much any radiation you can't specifically attibute to particular stars. For instance, the infrared background

12:08 AM

@ACuriousMind "Any kind of radiation"?

Blackbody radiation?

Hawking radiation?

Physics Guy

No

Cosmic rays?

user218912

he said any.

Physics Guy

Hawking radiation is special

@0celo7 Start with the Radon measure on the $C_c$ functions, define its extension to the Baire functions (perhaps those are what you call semi-continuous, they are basically functions that can be well-approximated from below by sequences of $C_c$ functions), then divide out the null functions

12:13 AM

Jesus

What is the first Radon measure supposed to be?

It's given to you, it's the input to the process

If you want to get the Lebesgue integral, you might take e.g. the Riemann integral

Ok, so we're doing the same thing.

@SirCumference Yes, what is unclear about that expression?

Your Baire functions are my semicontinuous functions.

But you remember all the details of that stuff?

@ACuriousMind So all of the examples I listed counted as CBR?

12:15 AM

No, of course not, I rarely remember the details of anything

You just have to know where they are to look them up

Hmm.

@SirCumference Yes, I'd say so.

LAB REPORT TIME

12:46 AM

@ACuriousMind Do you know any counterexamples to Dini?

@0celo7 I don't know what Dini is, and I don't know how one could know counterexamples to a theorem, either. (You probably mean examples where one of the hypotheses and the conclusion are untrue)

@ACuriousMind Dini is that monotone pointwise convergence of continuous functions on a compact metric space to a continuous function implies uniform convergence.

I want to know what happens if the limiting function is not continuous (because clearly the convergence is no longer uniform)

What about the standard example of pointwise convergence against a non-continuous function, $x^n$ on $[0,1]$?

Nice, yeah.

That's monotone

danke

1:10 AM

@ACuriousMind how the heck does $\liminf$ work?

I have a function $f:X\to\Bbb R\cup\{\infty\}$, $X$ metric

And it's lower semicontinuous at $x\in X$

So for $c<f(x)$ there is a nbhd $U$ of $x$ s.y. $c<f(y)$ for all $y\in U$.

Now, let $x_n\to x$ be a sequence, then it's clear $f(x_n)>c$ eventually for $n$ large enough.

This is possible for all $c<f(x)$, so somehow $\liminf f(x_n)\ge f(x)$?

I know that $\liminf f(x_n)=\lim_{n\to\infty} \inf_{m\ge n}f(x_m)$

But how do I actually use that definition?

I guess $\inf_{m\ge n}f(x_m)\ge c$ or something?

So like, let $c_n$ be a sequence s.t. $c_n\nearrow f(x)$

2:00 AM

@0celo, can you help me out with some QM?

yes

Our astronomy class suddenly jumped from basic orbital mechanics to QM. I gotta understand degeneracy pressure vs. thermal pressure.

I know the former is caused by the Heisenberg uncertainty principle, but why would confining an electron increase its kinetic energy?

if you decrease $\Delta x$, you increase $\Delta p=m\Delta v$

so $\Delta v$ goes crazy, and the KE goes up

If $\Delta v$ increases, shouldn't KE just become less predictable?

Since we're increasing the uncertainty, right?

Ok, you're gonna be tricky. You have to use the definition of standard deviation

2:05 AM

The mean of the sum of all distances from the mean?

Very horribly put by me

So we have $\Delta v^2=\langle v^2\rangle -\langle v\rangle^2$

that's what variance means, right?

@0celo7 Isn't variance just the root of SD?

where $\langle\cdot\rangle$ is the average

@SirCumference other way around.

@0celo7 Yeah, sorry

so the SD is $\sqrt{\langle v^2\rangle-\langle v\rangle^2}$

2:07 AM

All right, evidently I need to ask a basic question. What are the brackets indicating?

average

K

So do you agree that $\Delta v^2\le \langle v^2\rangle $?

Now I'm assuming $\Delta v^2 = ⟨v^2⟩ - ⟨v⟩^2$ is true because of something in QM I haven't learned?

The teacher spoke at a horrendously fast speed, I might have missed it

what does $\Delta v^2$ mean if not that?

2:09 AM

@0celo7 Good question. Wished he explained it.

this is high school AP stat

Do you want me to explain that?

@0celo7 Never took stat in HS.

ok let's review random variables

Only learning it now in college

$S$ is a set (sample space), $X:S\to\Bbb R$ a random variable.

Then $\langle X\rangle=\mu_X$ is the mean of the variable, given by $\int_\Bbb R xf_X(x)\, dx$ where $f_X(x)$ is the pdf.

The variance is $V_X=\langle (X-\mu_X)^2\rangle$.

It can be shown (by you) that this equals $\langle X^2\rangle -\mu_X^2$.

user116211

2:12 AM

Aha! So, you are teaching moments.

yes

the variance is the second moment about $\mu_X$

user116211

yes.

then $\sigma_X=\sqrt{V_X}$ is the standard deviation

in QM we write that as $\Delta X$

@SirCumference Clear?

@0celo7 Hold on

@0celo7 PDF?

Probability distribution function.

Defined as $f_X(x)=F_X'(x)$, where $F_X$ is the cdf

$F_X(x)=P(X\le x)$.

2:15 AM

@0celo7 I'm guessing "cdf" means cumulative distribution function

Yes.

user116211

$\mu_2 = {\mu_2}^\prime - {{\mu_1}^\prime}^2$

wtf?

user116211

it's variance.

Ok, so $\langle v^2\rangle\ge \Delta v^2$, and we're making $\Delta v^2$ large

the KE is $\frac{1}{2}m\langle v^2\rangle$

2:17 AM

I'm still memorizing all the logical quantifiers...

I don't see any logical quantifiers above.

user116211

^^

Does "→" count?

No.

that's an "arrow"

Well clearly I need to redefine a quantifier...

2:18 AM

arrows are common in e.g. wild west movies

@0celo7 ....

You know what I'm saying

I guess so...

Set language

user116211

Read Jech.

@MAFIA36790 I've encountered them all before. This is coming back to me

2:19 AM

@MAFIA36790 stop trolling

user116211

I'm not trolling :(

no physicist needs to read Jech

hell, no mathematician needs to

user116211

._.

@0celo7 So $\Delta X$ is simply the standard deviation of the position?

@SirCumference $X$ is just a general random variable

2:21 AM

@0celo7 In the case of the Heisenberg uncertainty principle

then you'd put in $x$ or $p$

So delta refers to the STD?

user116211

@SirCumference STD??

gross

Crap

I was thinking STandard Deviation...

user116211

2:22 AM

SD!!

I know!

user116211

not STD ;/

@SirCumference Exercise: show that a particle of mass $m$ in a spherical volume $V$ has KE $$\langle \epsilon\rangle \ge\frac{4.87\hbar^2}{mV^{2/3}}.$$

@0celo7 Let me remember this, we even went over something similar in class...

Actually I think $=$ holds if I were to state the problem more carefully.

2:25 AM

@0celo7 You're giving me that equation to work with?

If you assume the wave function is symmetric about the origin.

@SirCumference You are to prove it.

@0celo7 Yeah, just looked at the $ϵ$...

user116211

What? The stat class is finished :(

user116211

You were doing QM, ohh.

screw math, this is a physics chat

2:26 AM

All right, we know $\Delta X \Delta p ≥ \hbar$

This chat has been overrun by mathematicians lately. We need to reclaim it for Feynman

user116211

I learned bilinear transformation of random variables yesterday.

@SirCumference that's hardly right

user116211

It was fun.

@0celo7 Yes, it's greater or equal to

Screwed up

user116211

2:27 AM

But it was a mess when finding the limits of the new random variables.

still wrong.

correct me?

no

Well then

I've got a lab report to do

bye

user116211

2:27 AM

$\hbar/2\,?$

...

user116211

@0celo7 o/

So wait

First let me clear this up

Is $\Delta X$ just the SD of X?

user116211

yes.

proof?

2:30 AM

@0celo7 ...

I thought that was the definition?

sorry I'm a troll

carry on

All right

dmckee

@SirCumference I'd like to second what @rob said. College is one of relatively few chances you're going to get to try things out just because it seems like a good idea. For most people the next time that happens is at empty-nest time (circa 25-30 years down the road!).

Go for it while you have the chance.

user116211

@DanielSank o/

\o

2:34 AM

Wow, who starred that?

@0celo7 Finally caught.

user116211

You should write an analysis book; I'm serious @0celo7.

why @MAFIA36790 ?

user116211

Becuase you write well.

@0celo7 Ugh, could you show me where to start with the proof?

user116211

2:35 AM

Anyways, I'm back with my Born-Wolf.

user116211

I'm doing optics!!

@DanielSank I don't know what you mean.

@0celo7 He means this

@SirCumference ^

Oct 11 at 15:57, by 0celo7

Calling someone a troll is a violation of policy.

Oct 1 at 1:58, by 0celo7

@DanielSank I don't troll.

2:36 AM

last $=$ should be a $\ge$

Jul 5 at 2:11, by 0celo7

I'm not a troll

yeah, I'm not a troll

Jun 22 at 14:42, by 0celo7

@Danu I'm not a troll...

what are you on about?

Mar 14 at 0:27, by 0celo7

I'm not a troll.

2:37 AM

6 mins ago, by 0celo7

sorry I'm a troll

@DanielSank slander

Feb 17 at 4:43, by 0celo7

@vzn I don't troll.

I never said that

Hahaha

Feb 9 at 2:55, by 0celo7

@DanielSank I don't troll.

2:37 AM

If I'm a troll, then ban me

OK that's enough

@0celo7 I seriously don't follow

Nothing you mentioned implied the first equation

what's the first equation?

@SirCumference the $\langle x\rangle =0$ one?

@0celo7 $\Delta x^2 = \langle x^2 \rangle = \Delta y^2 = \langle y^2 \rangle = \Delta z^2 = \langle z^2 \rangle = \frac{1}{3} \langle r^2 \rangle$

which part is confusing?

Where that was derived

Nothing you mentioned seemed to imply it

2:44 AM

hmm.

did you read the very first equation?

"Very first"?

do you know why $\langle x\rangle =0$ and the others too?

Afraid not.

what does the line say?

write it here

I need to make sure you're reading

@0celo7 "The sphere and the probability distribution have both inversion and rotation symmetry."

2:48 AM

Yes!

So what do we expect happens to $\langle \mathbf x\rangle $ as $\mathbf x\mapsto R\mathbf x$, $R\in\mathrm{O}(3)$?

@0celo7 Sigh...can you use english?

$\mathrm{O}(3)$ = rotations and inversions

I don't recall basic set theory as well as I should

basic set theory?

this is calc 3

Well, I recall ∈

I only just finished calc 2...

2:50 AM

@DanielSank are you here?

This is bad...

what point in $\Bbb R^3$ is invariant under all rotations and inversions about the origin?

@0celo7 I'd suppose 0

yes!

That's easy enough

2:51 AM

So because everything is symmetric, we expect $\langle \mathbf x\rangle $ to be rotation and inversion invariant

but what is the only vector that is invariant?

@0celo7 0...

Yes, now it makes sense

ok, so keep working

come up with a smart question if you get stuck

@0celo7 no

@DanielSank lies

i bet if Bernardo were here you'd help him

Well, now we're at the next problem. If $\langle x \rangle = \langle y \rangle = \langle z \rangle = 0$, then how did we get the second equation?

2:55 AM

@0celo7 Do you need help?

in a general sense, yes

but I was referring to @SirCumference

@0celo7 I'm not qualified to give general help.

I have a degree in physics.

Does someone need physics help?

@DanielSank I vow to never do math in this chat again.

It's a little ridiculous what's been going on in here

@0celo7 Ok. Is that related to someone needing help?

No

@SirCumference um

I KNOW I told you what $\Delta x$ means.

2:57 AM

@0celo7 Yes

It's the SD of x

which is given by?

the fomula

$\sigma_x$ is a considerably better symbol for that.

^

@0celo7 Well obviously the root of the variance

dude

3:37 AM

@obe stop changing your damn name

I had to try 5 times to get it right

@SirCumference do you understand it?

user228700

Hi everyone :-)

@Kaumudi How goes it?

user228700

@0celo7: It's going OK. U?

doing a lab report

not very happy

user228700

Oh :/ Well, good luck!

user218912

3:43 AM

@0celo7 i won't change it anymore...

user218912

for real

Proof?

user218912

only time will tell

If it doesn't change for 2 years I will get you HE.

user218912

1st edition

3:45 AM

no

if you guess which artist I am listening to right now I will get you HE 1st ed

user218912

khaled?

You could have picked the one Canadian rapper

user218912

:(

@obe can you come up with my top 5 rapper list?

user218912

hmm

user218912

3:57 AM

I don't even know 5 rappers.

Surely you must

Tupac, Drake, Future, Dr. Dre, Jay Z

not my top 5

but you certainly must know them

Snoop Dogg, Lil Wayne

user218912

yup I know them

Kanye

user218912

okay let's guess