8:18 PM
@0celo7 I'm a bit meh'd about the Cantor set at the moment
but if you do find a non-Jordan-measurable open set that'd be pretty interesting

8:31 PM
@EmilioPisanty I got u

@0celo7 nice

@EmilioPisanty I was playing around with Cantor sets and got this
@EmilioPisanty Next exercise is to show if it's sigma additive or not
It's not in the strict sense since the Jordan measurable sets are not a sigma algebra

@0celo7 what is?
Jordan measurable sets?

Jordan measure

as in, if a disjoint union is still measurable, the measure is the sum of the components?

8:43 PM
@EmilioPisanty no, those aren't. But assuming I have a sequence of disjoint measurable sets, is the measure of the union just the sum of the measures of the sets

man, that sounds so boring
Used to be I found that interesting
it appears that it's no longer the case =/

It's true if you use Lebesgue measure. I think Jordan measurable sets have the same Lebesgue measure

why spend so much effort on a doomed, doomed, doomed concept of measure?

So just use sigma additivity of that one
@EmilioPisanty fun

@0celo7 surely there's better places where you can have that fun?
go do some functional analysis instead

8:45 PM
I like functional analysis too

or go do drugs like normal people

ugh
@AccidentalFourierTransform never say things in the h bar that you're not comfortable with getting starred =P

@EmilioPisanty if I had my way right now I'd be playing MGSV

@EmilioPisanty I regret nothing
for example
I love ISIS!!!
(the metal band, of course)

lolz

8:51 PM
but that is SO

a factor of just under two between term time and holidays
@AccidentalFourierTransform ...so what?
=P

it would be nice to have the info about PSE only
no, I mean, that graph is certainly interesting

don't think the analysis exists
also

but, yeah, Id like to know about PSE

good to see the home team topping the UK charts

8:52 PM
why only English-speaking countries though?

@AccidentalFourierTransform 'cause the SE analysis team are sissies

I am very curious what people think of this:
in Physics Meta, 47 mins ago, by Qmechanic
@DanielSank : If after all this discussion about homework in the past, the rep. king of Phys.SE still doesn't use politically correct words for homework, we can't expect to educate other users as well. Also note that the close reason already says homework-like (rather than homework).

can't be bothered to brave another language

Does a user's rep indicate that their behavior, in particular word choice in discussing site policy, should be more highly valued or more indicative of site direction than other users?

man, I don't think I make half that number
yeah, I don't

8:54 PM
@EmilioPisanty Yes, and I have the highest fraction of accepted answers for people who have written at least ~50 answers.

@DanielSank well, fancy being you, then

It's my claim to fame.

I should get off my ass and award more rep in bounties than you currently have =P

...and when someone turns to JR's rep as an indicator that his behavior should somehow specially influence site policy, I am not going to be shy about sharing my claim to fame ;-)

@DanielSank let me play devil's advocate here: maybe QM meant that JR is one of the most active users (regardless of his rep), and as such, his behaviour may be a good representative of the general comunity

8:56 PM
@EmilioPisanty Wait what?
@AccidentalFourierTransform He's one guy though. Having a lot of use does not mean you represent the community. In fact, it's probably the opposite since most users aren't power users like JR.
...Mr. Devil.

@DanielSank E. is the user who's awarded more bounties here in PSE
by far

@DanielSank the total of rep I have awarded in bounties is 70% of your total rep

@EmilioPisanty That's amazing, but how is it related to previous discussion?

would take a bit, but not too much, of sustained effort to bring that up to 100%

@EmilioPisanty -_- jeez
Where U get all dis rep?

8:58 PM
@DanielSank no idea

@EmilioPisanty ohhhhh
So your claim to fame is that you've given away the most rep? That's awesome!
You're a regular Robin Hood.

@DanielSank pretty much, yeah

That's a pretty sick claim to fame.

@DanielSank don't think I ever stole any rep

@EmilioPisanty Hmmm, ok then you're just a philanthropist.
In any case, pretty awesome.

9:02 PM
@DanielSank philantropy is when it actually costs you
giving bounties above 30,000 costs you nothing

@EmilioPisanty so there is no word to describe you. Congratulations!
(there probably is some German word, though)

I just like to think of it as doing my bit to improve the site by putting the rep incentives where they will produce the most quality content

@EmilioPisanty It's a really nice idea.

you are such a romantic

Thank you again for rewarding that post about the Boltzmann constant. It was nice to see that appreciated.

9:04 PM
@DanielSank it was a really good answer

@EmilioPisanty Thanks. Pretty sure the first place I encountered the idea that temperature is a Lagrange multiplier was in Pathria's book.

and also, I think the site benefits from having folks who have produced that sort of high-quality content to reach the important moderation milestones faster

I'm surprised, in retrospect, that Reif doesn't discuss that.
@EmilioPisanty Perhaps. I feel I don't exercise my moderation powers much at all.
i.e. I never do reviews any more.
Like, ever.

@DanielSank maybe not, but you're able to delve into the full review queue history and suss out bad stuff if the need comes
also see & react to deleted answers, again, if there's a need to
It would be really good to have a much bigger critical mass of 10k+ users, as measured by the fraction that do occasional moderation
particularly if we as a community managed to actually use things like the recently-deleted list on the 10k tools
actually catch bad deletions if and when they happen

See, that sounds like work.

9:09 PM
@DanielSank Whoa! I hadn't seen it formulated that way. Did you do a post on it? Pointer?

I just get so... I dunno... funky... when I do reviews.

@DanielSank yeah, the 10k tools could use a fair bit of improvement

42

We can understand all of this business if we visit the statistical mechanics notion of temperature, and then connect it to experimental realities. Temperature is a Lagrange multiplier (and should have dimensions of energy) First we consider the statistical mechanics way of defining temperature....

@dmckee
This is the right way to think of temperature. Statistical mechanics is a maximization problem where you're maximizing entropy.

also, on the bounty-to-10k front, here's a good morsel of history
3

Suppose that for all $z$ in some open set $Z$ of complex numbers containing $z_0$, the Hamiltonian $H(z)$ is a compact perturbation of the self-adjoint $H(z_0)$ depending analytically on $z$. Then, for every simple eigenvalue $E_0$ of $H(z_0)$ and associated normalized eigenstate $\psi_0$, there ...

But! You have constraints: fixed energy, fixed volume, fixed particle number...

9:11 PM
that bounty put Arnold Neumaier over 10k

Temperature, pressure, and chemical potential are the Lagrange multipliers associated to fixed energy, pressure, and particle number!

In a couple of days I'll put heather over 5k :-)

@EmilioPisanty You like putting people over 10k, eh?

"Now suppose we add the constraint that the system has a certain amount of energy [...]" <== Sets off one of those "How did I miss that?!?" moments right at the top of the post.

@DanielSank have done ever since

9:12 PM
@dmckee I know, right?!

Some days I feel like a dunce.

@dmckee Don't. Very few physicists seem to have ever seen this idea.
I just got lucky that I read Pathria and picked up on that little note.

@DanielSank I once saw a public lecture by Eric Cornell that I think you would have enjoyed immensely
Core point that nature is lazy and sloppy
so it likes to maximize the sloppiness and the unlaziness

@EmilioPisanty I had the privilege of eating dinner with him once. Friendly guy!

@DanielSank this was at ICOLS 2013, and all the other talks were posted online. I emailed after a few months to ask, and they said that he's so perfectionist that he didn't want it aired until it was absolutely spotless.

9:15 PM
They have it in our library here. I'll be back.

so nature likes to maximize sloppiness S and minimize unlaziness U

@dmckee What's "it"?
Cornell's talk?

but obviously that's a problem because being really sloppy also requires you to be very unlazy, and being unlazy isn't fun
so nature tries to minimize some kind of figure-of-merit
that goes like U-S

Free energy?

@DanielSank He probably means Parthia.

9:17 PM
but obviously that's a bit too rigid

Forgive autocorrect

so you need some kind of tradeoff factor
so you minimize F=U-TS

@0celo7 Oh right.
@EmilioPisanty Right. So here's why I like the Lagrange multiplier picture so much:
The value of a Lagrange multiplier tells you how much better you could maximize your objective function if only you would be willing to relax the constraint a bit.
A system with high temperature is one where giving it just a tiny bit more energy would let it increase the entropy by a relatively small fraction.
So, if you hook it up to a low temperature system, it will give up energy, because the low temperature guy can raise the total entropy a lot.
By the way, this helps understand why negative temperature means something is very hot.
If $T<0$, then I always want to give away energy.

damn

What?

9:22 PM
one of these days I'll really truly grok these things

@EmilioPisanty It's not complicated. I promise.
If $T<0$ that means that giving away energy raises $S$ for me.
If $T>0$ then absorbing energy raises $S$ for me.

oh, no, I understand everything you said

Therefore, if a negative temperature system touches a positive temperature one, the energy flow is from the negative guy to the positive guy.

I just would have been entirely unable to come up with more than 10% of it on my own, I should think

@EmilioPisanty Ah well, we all stands on the shoulders of our predecessors.

9:24 PM
12 mins ago, by dmckee
"Now suppose we add the constraint that the system has a certain amount of energy [...]" <== Sets off one of those "How did I miss that?!?" moments right at the top of the post.
again.

Interesting.

I'm still waiting for the Eureka moment

When I read it in Pathria my reaction was more like "Oh! Well why didn't my undergrad course just say that?" along with "Man, Lagrange multipliers are way more interesting and important than I realized when I first learned about them".
3

hello all

Hi!

9:39 PM
oh, i noticed the homework comments made here in chat - I'd certainly be willing to help in any way I could.

@DanielSank I had to feel very comfortable with the basic framework of Lagrangian mechanics before I could focus on the undetermined multipliers problem deeply enough to understand it and start to see the depth of it.

Howdy

That didn't happen the first time through for me either.

@dmckee Ack! You can understand Lagrange multipliers without ever thinking about Lagrangian mechanics!

That stuff is like magic.

9:47 PM
The basic idea is rather simple!
No no no no. There's no magic. I promise.
Suppose you have a function $f(x,y)$ that you want to maximize.
How do you do it (real question, not rhetorical).

@DanielSank Formally. Identify generalize coordinate. Write the energies. Take derivatives. Write constraints. Take derivatives. Setup system of equations. Beat at the math until one of us surrenders.

@dmckee I suspect you'll enjoy this one

Is there a better way?

@dmckee Woah, dude. I'm just saying you have a normal, friendly function of two variables and you want to find a local maximum.
You just compute $\partial f / \partial x$ and $\partial f / \partial y$, set them to zero, and solve.
Right?
No physics here.
No magic.

@DanielSank Oh. Yeah.

9:50 PM
Friendly indeed

@dmckee Good ok. Now suppose I ask you to find the maximum constrained to the condition that some other function is equal to zero, e.g. $g(x,y)=0$.
Now we have a hard problem because setting partial derivatives of $f$ to zero will find us points that aren't on the constraint. Ok so far?

Parameterize the solution to $g=0$ and take the derivative along that line, I guess.

@dmckee That would absolutely work. Unfortunately, finding a parametrization of a curve defined by a (set of) equation(s) is very hard in general.

And cry is the solution isn't well behaved.

@dmckee Well let's see if we can avoid tears, eh?

9:52 PM
also this

@dmckee The equation $g(x,y)=0$ defines a curve in the $x/y$ plane.

@DanielSank Sure, but like you said it is probably hard to parameterize.

@dmckee Yes, but the curve is there.

And it might be a more complicated set than a curve, but lets pretend that doesn't happen in physics.

Now here's the trick: The curve is always perpendicular to the gradient of $g$.
It has to be, because if it weren't, it wouldn't be a curve of constant $g$!

9:55 PM
@DanielSank Hmm ... I think this is the clever part.

Let me know when you agree with that, or if you have a question there.

No question, that's clear.

Ok cool.

@DanielSank objection

Now suppose we're walking on the curve, and as we walk on it, our value of $f$ is changing.

9:56 PM
@dmckee unless you have a cusp, the implicit function theorem does give you a curve...

you never specified an inner product in $x,y$ space

If that's the case, we're not at a local constrained extremum.
Ok?
@EmilioPisanty Oh do shut up ;-P
This will help:
@dmckee I await your confirmation or objection.

@DanielSank Yeah. That's why I wanted to take the deriviative along the curve.

@dmckee Great. But now observe: At the point where we're at a local constrained extremum, the curve of constant $g$ must be parallel to the curves of constant $f$.
Note the picture.

Yeah. Got it.

9:58 PM
Ok then we're on the home stretch! We said the gradient of $g$ is perpendicular to the curve. We also know that the gradient of $f$ is perpendicular to the curves of constant $f$.

@DanielSank ah, yes, I remember that

Therefore, at our constrained extremum, we have $$\nabla f || \nabla g \, .$$

Cute.

Or, in other words, $$\nabla f = \mathcal{L} \nabla g$$ where $\mathcal{L}$ is just some constant.
So, $$\nabla ( f - \mathcal{L}g ) = 0 \, .$$
In other words, we now have an unconstrained maximization problem for a new function $f - \mathcal{L}g$.
Done.
That's Lagrange multipliers.

@DanielSank so how does your previous comment look like in this picture?

10:01 PM
Yeah. If they ever showed me that in a context outside of Lagrangian mechanics I forgot. Or was on the beach. Or something.

42 mins ago, by DanielSank
The value of a Lagrange multiplier tells you how much better you could maximize your objective function if only you would be willing to relax the constraint a bit.

@dmckee Beaches are good.
@EmilioPisanty I was just about to get to that.

I'll just go order my high-durability dunce's cap, now.

@DanielSank ok cool I'll shut up

@dmckee In mathy terms, if $r(t)$ parameterizes the solution set of $g=0$, then $t\mapsto f(r(t))$ must have a local max. By the chain rule, we must have $\nabla f\cdot r'=0$. Then note that $r'\bot \nabla g$, and you're done.

10:02 PM
Suppose the point at which we find this constrained extremum is at a point where $f$ is very flat.
In fact, suppose $f$ is perfectly flat.
In that case, moving the constraint curve a bit doesn't help us at all, i.e. there's no gain in $f$ if we wiggle the curve.
@EmilioPisanty Ok?

@DanielSank Can you make the same thing work for maximization on spaces other than $\Bbb R^n$?

@0celo7 Later.

@DanielSank perfectly flat, yes

@EmilioPisanty Yeah, well if $f$ is pretty flat then $\nabla f \approx 0$.

for $f$ pretty flat, I'm not 100% sold
it won't change much, sure

10:04 PM
We have $$\nabla g = \nabla f / \mathcal{L}$$

this is (only) at the extremum, right?

Yes.

ok sure

$\nabla g$ is some fixed thing, so if $\nabla f$ is small, then we need $\mathcal{L}$ to also be small for the equation to work out.
In other words, small $\mathcal{L}$ corresponds to the case where your're in a flat area of $f$ and relaxing the constraint a bit doesn't help change the extremized value of $f$ much.

@DanielSank sure

10:06 PM
Large $\mathcal{L}$ corresponds to a case where "the constraint is holding you back a lot".
So, @dmckee, in the case of Lagrangian mechanics a large Lagrange multiplier means that the action would be much more minimized if the constraints were relaxed. In other words, the constraint forces are large.

@DanielSank Nice.

@dmckee See, no magic!
Unfortunately, this entire topic is usually taught by dumping a problem on the student and telling them to follow some god-forsaken recipe to get an answer.
When I retire, I would like to write a book titled How to understand all that stuff they never explained.

you should
you explain things very well.

Yeah... chapters: Entropy, Lagrange multipliers, Fourier transform (already written)... what else?
Maybe I should throw in a cooking chapter. That would be awesome.
Oh, jeez, tensors.
My goodness yes there will be a chapter on tensors.

YES

10:12 PM
Yes, and with pictures.

i still don't really understand those buggers.

Hi there @heather

@AccidentalFourierTransform hello =)

people didnt seem to like some (or most) of my questions, because I got many downvotes
it was probably because of the last question
which was a joke, but some people didnt realise

@AccidentalFourierTransform I down-voted it simply because it asked more than one question.

10:14 PM
(or they did but disliked it anyway)

The questions were individually fine.

@DanielSank thats what I thought

@DanielSank sobolev spaces on manifolds

@heather take it easy with the tensors

therefore, I deleted all but the first question

10:15 PM
they look big and scary but they're honestly not that necessary for a pretty long while

@AccidentalFourierTransform Oh, I should un-downvote.

@DanielSank OF COURSE YOU SHOULD

@AccidentalFourierTransform well, anyway, maybe you could post the other questions as separate posts? I thought at the very least that #2 and #3 were good.

yeah, I will post them again
I wasnt very motivated to do so because of the several downvotes
ppl didnt like my questions :-(
but yeah, Ill post them again

10:18 PM
=/ i thought they were good. and #4 was kind of fun.

@AccidentalFourierTransform Oh stop moping and post them separately.
I liked them all.

-7

Einstein or Bohr? Harvard or MIT? Michelle or Melania? Tesla or Edison? Ubuntu or Fedora? Heisenberg or Schrödinger? Dems or Repubs? Strings or loops? Socialism or Capitalism? Frequency or time? Trump or Hitler? David Z or Ron Maimon?

^-7!? geesh.
i didn't think some of them were too bad.

@heather It's too many questions all together!

well, yes, i suppose that is problematic.

...and several of them are ridiculous pop culture questions with zero context.

10:20 PM
also true.

Michelle or Melania wat?

i think that references the first ladies.

Yes but what's the question?
Which one you'd rather hang out with?

but really that's not a fair comparison - Mrs. Obama had 8 years, Mrs. Trump had a few weeks.

Which one is a better first lady?

10:21 PM
@DanielSank which one's better, I think.

Which one has better taste in polar bears?

lol
definitely that.

which one embodies the cultural values you feel closer to

Well then it's Melania because she's got the ties to Russia.
::ducks::

like the rest of the questions

10:22 PM
wow

I thought "Einstein or Bohr? [...] Heisenberg or Schrodinger? [...] Strings or Loops?" were the best ones.

Let's see if I get suspended for that

because they asked more viewpoint/physics based questions.
hmm...
0

Does Level IV Multiverse/Ultimate Multiverse contains 'impossible worlds'? Does it contain universes with sets, structures, or systems that exist beyond spacetime, duality, or existence and nonexistence? Does it contains universes with different laws of logic or metaphysics than ours? Does it con...

@heather Because people don't understand how to VTC properly.

but John Rennie!

10:24 PM
It should be closed for asking too many questions because that conveys useful information to the poster.
@heather But John Rennie what? Clearly you didn't see my little rant a bit up...

no, I didn't.

John has a lot of rep because he writes a lot of answers. That doesn't suggest that he is better at site policy than anyone else.

@DanielSank i think they're all very closely related and basically asking the same sort of thing. I don't think it should be closed at all.

@heather Look here

10:25 PM
@DanielSank what? it doesn't ask too many questions! It only asks one question!

@DanielSank he's more familiar with the site, he's been around for a while, and he's one of the most respected people on PSE. I count that as suggesting that he's better at site policy.
@DanielSank now that comment is preposterous.

@heather Which?

@heather He is not "more familiar with the site"!

he's been around for a while!

10:27 PM
When the mod election happened JR showed several times that he does not understand how large parts of the site work! I'm not saying he's more ignorant than anyone else. I know little about site workings as well.
I just wish everyone would stop equating rep with "knowing about site policy" because it's wrong.

@DanielSank for the record, that's not the reason I think JR knows more about site policy.

@heather I'm not going on chat safari right now, sorry.

@DanielSank I don't think Qmechanic's comment should be read quite like that. Make a reasonable switch to "if someone who uses the site as often and for as long as JR still gets X wrong".

sorry, but a claim like that i want a reference for @DanielSank.
::shrugs::

@EmilioPisanty But that's one person. And besides, the entire point is sunk by the fact that the policy has been called the "homework" policy and not something else.
There's no reason to expect anyone to call it anything other than the homework policy as of yet!
@heather That's reasonable. I might go hunting later, but I doubt it because I don't see the point in question as particularly important.

10:31 PM
@DanielSank ::shrug::

Why is everyone shrugging at me?

cf. JR's comment on the right
6 hours ago, by John Rennie
@DanielSank I gave up debating the homework policy in favour of hammering nails into my kneecaps because it was less painful and more productive.

@DanielSank okay. I am interested, so if you find a comment, send it my way =)

it takes a good bit of energy to engage with that debate at this point

10:43 PM
Apparently, I live in Atlanta

ah, darn it
my motivation to answer the question is now way down.

@AccidentalFourierTransform good trap music there

10:59 PM
0

In a computer simulation, what math formula do I use to predict what will happen when a cylinder hits a triangle's point if both objects were made up of pure titanium? In my mind, I picture the cylinder will have a dent in the shape of a cone assuming the cone is traveling at 100mph and the tri...

trap music... shoo...

^@mod, do you want to invite this person

@heather she has 140 rep points in SO
she can access the chat

huh, so she does
> This helps alot! Thanks!
^warm fuzzies
This helps alot! Thanks! — science error 1 min ago
interesting.
(sorry, just experimenting there)

well g'night for me
bye peeps

11:06 PM
Cya.

11:17 PM
good night @AccidentalFourierTransform

How goes the AMA prep?

okay, i suppose. i don't know that there's much i need to prep now =)

Have you set a time?

yup
Monday, February 20th, at 10pm UTC
the question pool post is here

Nice.
Have fun :-)

11:27 PM
i'm looking forward to it =)

@AccidentalFourierTransform shoo?
How can you be from ATL but ain't bout that trap life

ATL will live in shame forever after that Super Bowl.
Dunno which is worse, Buffalo's back-to-back loses or Atlanta giving up a 28-3 third quarter lead.

11:56 PM