@BalarkaSen Ok, so I have to integrate by parts for this to work. Recall Green's theorem $$\int_M (h\Delta f+\langle\nabla h,\nabla f\rangle)\, dv=\int_{\partial M} h \, \nu f\,d\sigma,$$ which holds for $h\in C^1(\bar M)$ and $f\in C^2(\bar M)$.

So if $\nu f=0$, like I am hoping, then $$B_f(h)=\int_M(h\Delta f+\langle \nabla h,\nabla f\rangle)\,dv=0$$ for any $h\in C^1(\bar M)$. Assuming $M$ to be compact, we find that $B_f$ is bounded. But $C^1(\bar M)$ is dense in $H^1(M)$, so $B_f$ extends to all of $H^1$. Also, its $H^1$ operator norm is zero...so it is the zero operator. This means I can integrate by parts even with Sobolev functions!

I think my program can work, but there are still more details!

@BalarkaSen The proof I had in mind was actually ridiculously complicated. Hint for a better one: Let $\mathfrak{iso}(M,g)$ denote the space of Killing vector fields on $(M,g)$. (i) Show that $\mathfrak{iso}$ is a Lie algebra. (ii) Show that for any $p\in M$, the map $\mathfrak{iso}\ni X\mapsto (X_p,\nabla X_p)$ is linear and injective.

@heather part of a larger system

ah, okay.

thank you.

@HDE226868 What about central pressure?

It might make more sense for me to use the pressure from an analytic perspective

@0celo7 Will you smite me if I give it to you in cgs units?

Try something on the order of $\sim10^{24}\text{ dynes cm}^{-2}$.

That figure I gave you for density was towards the upper end of the spectrum; so too is this.

@0celo7 I should use Mathematica more often.

@HDE226868 Not me. I'm open minded.

@dmckee Good to know.

It took me two years to start thinking in SI units, as opposed to imperial.

Not sure how long it'll be 'til cgs becomes a bit more normal.

@HDE226868 what the hell is a dyne

@HDE226868 Ideally I would have it in $G=c=\hbar=1$ units

@0celo7 I've said too much already.

$10^{-5}\text{ N}$, though.

@0celo7 Not doing that conversion for you.

@0celo7 I need to know if the cosmic event horizon's distance changes in a universe with $w = 2/3$

Or alternatively if the scale factor increases exponentially

@HDE226868 it would probably be easier to just put in the cs and Gs into my code

@0celo7 So are you doing a full relativistic treatment?

@HDE226868 of course

@JohnRennie After several hours I got a crash on the same map in the same spot (ish)!

@HDE226868 any idea for $K$ in $P=K\rho^\gamma$?

@HDE226868 Oh, guess i should just plug in the pressure and density you gave me.

@JohnRennie quick, what is the radius of a star in meters

on average

also good morning

$10^8$

@JohnRennie wow, I was several orders of magnitude off :)

Well it depends on what type of star. The Sun is about $7 \times 10^8$m.

@JohnRennie Right now I'm just playing around with the stellar structure equations

I seem to be getting 3,000 but this plot is for ants

3km. That's about the radius of a stellar mass black hole.

the $y$ axis here is...uh

not sure what's going on there

@JohnRennie I just told it to do the numerical integration out to $r=3,000,000$

I think I broke MMA

It's only using 9% of my CPU...cmon son

@JohnRennie Is the MMA software just not that good at high performance computing?

I don't have that much experience of using Mathematica. I really only use it for tensor algebra.

@JohnRennie If you know of another program that's relatively easy to use and can do massive nonlinear ODE systems, I'm all ears.

Maybe the more engineering orientated apps. Matlab possibly?

@JohnRennie I might try that. Never did a system in Matlab.

mathworks.com/help/symbolic/…

@JohnRennie How long should I let this disaster of a Mathematica script run?

I can't comment. Is there a way to run it with a coarser granularity or some other way to speed it up at the expense of lower accuracy? If so I'd start with a quick low accuracy run and gradually increase the accuracy.

@JohnRennie I think I know what's happening.

The star radius is around 825,000 meters and I'm extrapolating to 3,000,000.

@0celo7 FFS, I practically gave you $K$ earlier

The ODE probably becomes horribly singular at the actual radius

;-;

@JohnRennie Know of any way for me to trace the value of the graph along the tail there?

the vertical axis is still messed up

Change the scale on the $y$ axis?

@JohnRennie Yeah, or log it or something

Hmm, no, it's still 10E14 there.

@JohnRennie The radius is 846km

this is a compact mofo star

Sounds like you've modelled a white dwarf ...

@JohnRennie computing mass now...

@JohnRennie are masses reported including binding energy or w/o?

I think the binding energy of even white dwarves is quite a small fraction of their total mass.

@JohnRennie what's your ballpark for the mass?

I've no idea. If it's a white dwarf then around 1% to 10% of the mass of the Sun I guess. Say $10^28$ kg.

I can't get it to do the damn integral

this might have to wait for another day

@0celo7 Neutron stars can be on the order of 10km

It should be clear why stellar astronomy is cool :P

Hell, neutron stars are often 2-3 times the mass of the Sun

rubs eyes

this is future me's problem

Your future's so bright you gotta wear shades.

@TheRaidersofLasVegas Uhhhh

@JohnRennie mathematica has this amazing ability to break itself even though the same shit worked a minute ago

Welcome back @Cows

@Kaumudi.H I cheated. I ate some ice cream to keep me alive until the meal. Well, it was National Ice Cream Day after all :-)

What flavour?

Rhubarb

I've never tried that flavour.

Does rhubarb grow in India? I guess there will be areas cool enough for it.

Yup.

It's a very traditional English dessert because it's well suited to the English climate. Go back a few decades and everyone used to grow rhubarb in their gardens. Even I had a rhubarb plant in my garden.

Don't like the flavour of it though.

Too sour.

It was this iceceam

It was ... OK. I probably won't buy it again.

@TheRaidersofLasVegas rhubarb contains a lot of oxalic acid, which makes it sharp. In fact the leaves contain so much oxalic acid that they are toxic. However you can put in lots of sugar to counteract the sourness.

I like maple walnut ice cream.

Mint choc chip is my favourite. And by quite a long way. But the rhubarb icecream was on offer at a reduced price and I was curious to try it.

Praise Jesus

I had an error

The error can be corrected now

@Kaumudi.H yes.

We all know John eats strange things.

Though it's normally served sweetened as a dessert.

Rhubarb crumble is very popular as a dessert in the UK.

Sandwiches for breakfast sounds good to me. Although I'm still feeling rather full from last night's meal so I'm not that entusiastic about food at the moment.

typical German breakfast

Hyderabadi gosht

@0celo7 what?

@TheRaidersofLasVegas what?

It was very good. They cook lamb very well. Tender, but not overcooked.

I have had the Hyderabadi gosht before, though I think I didn't have it last time.

If you ask the chef nicely they will allow you to choose the lamb and slaughter it at your table :-)

No :-)

It was an attempt at humour - probably a poor one.

Dang, I would go to that restaurant

@BalarkaSen

we should find a place like that

perfect edge level

@Kaumudi.H in the UK meat processing is heavily regulated because of scares like BSE. There's no way any restaurant would be allowed to do their own slaughtering.

@TheRaidersofLasVegas why were you at a restaurant with a monkey?

gotta sleep. PDE will have to wait for tomorrow

cheers, all

Goodnight

@Kaumudi.H sorry

I did.

Cya @0celo7

It was a Chinese restaurant.

You're in Chennai? The water supply is erratic in Chennai?

You hinted the family might move back to Kerala. Is that just some vague long term possibility?

@JohnRennie Sounds interesting. I have seen Chicken getting slaughtered. Never seen lamb slaughtering.

"Afraid so" - you're not in favour of the idea?

Is that closer to your school?

Chicken slaughtering is gruesome, though.

@Kaumudi.H sorry miscommunication

Will your father be retiring, or getting a job in Kerala?

@Kaumudi.H Quality of schools is overrated.

Quality of students is the important thing, no?

True dat.

@Kaumudi.H is that the same school you went to?

Duh... Students should study only from high School IMO. Let them enjoy their early childhood. Else, they would miss it.

::makes popcorn::

I believe curiosity can be generated on its own. I don't believe teachers have a huge role to play there.

Say, someone looks at the stars and wonders what it is all about. You don't need a teacher to tell you to look at the stars. I say that curiosity can be generated by everyday activities. You just need someone to patiently answer your questions.

which could be your parents, or teachers or elders, etc.

How old are they?

Hmm, this is actually the time they should be asking more questions

mornin

hi pal

@0celo7 i'd just burn my copy of Das Kapital to have a place like that

I noted your set of hints for that exercise.

@Sid Students should only study for high school International Math Olympiad? Damn, you have have some effed up syllabus in mind for us.

@BalarkaSen .... Damn. IMO=In my Opinion

Though, I would encourage anyone interested in Maths to go for Olympiads. They are loads of fun, if people are to be believed.

I know, I was just joking

Though, If students prepare for IMO, they don't have to study High school Maths anymore

It's not entirely true that IMO is universally fun for mathematics enthusiasts.

It's certainly true for math fanatics :P

Under what condition is a curve of the same "magnitude" as its range

That is, a finite interval is finite

So that $\gamma((0,1))$ doesn't fuck off to infinity

What kind?

No lettuce?

@Slereah This is epic

Those are squares @Slereah

@skullpatrol Not in the $\ell_\infty$ norm

Are those lines parallel? @Slereah

Depends on the metric

Specify the metric please.

@Avantgarde There was a game today(yesterday) at 23:30 UTC.

Cricket?

@Kaumudi.H If you find it interesting then go for it, though I'm not sure how useful it will be. Your course may be very different to the videos.

Always good to get a different perspective.

Crash courses are very "quick & dirty."

It's alright

CS is a snooze anyway

Not if you read Knuth's Art of Computer Programming.

Snooooooze

Have you read his Concrete Mathematics book?

I did not

I have read a fair bit of computation theory

It's alright but odds are good you're not gonna do that in a CS class

True.

@Slereah if you do a reasonably theoretical course in CS, that sort of stuff should be covered, at least in a bit of detail. Or, I would certainly hope so anyway

Knuth's textbook has unsolved research questions as exercises.

So does Visser

The last problem in Visser's book is "quanticize gravity"

I keep hearing ppl saying sth like "cosmology is black magic". Anyone can shed some light on this joke for me?

I do not know

Cosmology is fairly straightforward

anyway, I am thinking hard now for my future career paths.

I enjoy a lot building my understanding.

but doing research so far seems to me quite ...

You'd better enjoy building websites because there's a bigger job market for that

not as much as doing one wants to do

@Slereah how about AI?

I may start a business later

AI is fine, but you'd better be good at it

Oh and you'd better do AI for data analysis, because that's what people want

They want to know what brand of bananas people want

lol

And god help you if you can't use a database

@Slereah it seems a risky business...

I have uncles who are highly successful business men in China

not sure if I should ask them if they are willing to take me under their wings

probably a good starting point for doing business

Hello guys I would like to ask about something related to set theory . If I would say { (x=a1) or (x=a2) or (x=a3) } , this means its a set containing a1 and a2 and a3 (I guess) but what if I wanted to mean that it may contain a1 or a2 or a3 , here ("or" has the meaning it has in english and not in math; not summing up ) so if I cant put it this way , is there another logical connective for that?

Have mercy on me Carter

@0celo7 Yeah, it is (no telling if it will really come out next month though). I'm also not sure you'll like the game, but I guess since it's free it doesn't hurt to try.

@ZaKh The notation you're using is not one I've seen before. Typically $\{a_1, a_2, a_3\}$ would be used to designate a set containing those three items.

If I understand correctly, you're talking about a situation where you have a set whose contents are unknown to you, but the contents could be just $a_1$, or just $a_2$, or just $a_3$, right? Or could the set contain more than one of those elements? Or none?

Either way, you could designate the set itself with a variable, like $X$ (it's common to use capital letters for sets), and then you could even write $X \in \{\{a_1\}, \{a_2\}, \{a_3\}\}$ (yes, that is a set of sets) to show that $X$ is one of the three sets $\{a_1\}$, $\{a_2\}$, or $\{a_3\}$.

@Sid I was asleep then

Does anyone know what $\natural$ is for

@Slereah Muscially, it means that any prior $\flat$ or $\sharp$ is invalid for the note it precedes.

Ah alright

(It is also used to denote the completion of a spacetime)

Budic and Sachs define $\downarrow U = I^-(\{ x | \forall p \in U,\ x \ll p \})$

Isn't that just $I^-(I^-(U)) = I^-(U)$

Hm

folks, some tagging feedback requested

I just created the nmr and relaxation tags

I wanted nuclear-magnetic-resonance but it's one character too long

should we make nuclear-magnetic-resonanc and redirect it to nmr?

or something like that?

@EmilioPisanty Hm, I don't think a "chopped off" tag is a very good idea

Maybe SE will increase the tag length network-wide soon, as they did on Literature

@ACuriousMind it would only be a synonym, so nothing would actually get tagged with it

but it would allow you to start typing nuclear-magn... and be redirected to the right place

@ACuriousMind if they do, we can make a full-named tag and redirect everything there

@EmilioPisanty Hm, maybe just put "nuclear magnetic resonance" in the first sentence of nmr's tag wiki excerpt? I think the tagging box also searches the excerpts

@ACuriousMind doesn't seem to

also, if you add spaces in the tag box it separates it as a single tag

Hm, well, that's crap :P

@EmilioPisanty wow, we didn't already have a tag for it? That's weird. Anyway, we could also try nucl-magnetic-resonance which sort of gets the idea across

@DavidZ you want that as the main tag?

and then we really hope that SE gives us another character for tag names

or as a synonym redirecting to nmr?

@DavidZ Then people typing 'nuclear' quickly won't see it

If we're gonna cut-off a bit of that name, I guess Emilio's suggestion of chopping off the end is the best

maybe nuclear-magnet-resonance? =P

@EmilioPisanty I dunno... personally I tend to prefer the longer one being the main tag but I don't know if it matters

@DavidZ I agree with longer tags as main, so long as they don't have bits lopped off

@EmilioPisanty That works too. Actually we could have both. Pick one to be the main tag and make a bunch of synonyms for it to help searches.

but then having as many synonyms as possible redirecting to the main target does no harm

Now that I'm thinking about it, the one we'd want to be the main tag should be the one that will make the most sense to people reading the question or seeing it on the index page or the like, because the main tag is the one that will actually get displayed

@DavidZ yeah

or maybe magnetic-resonance as main?

with the others redirecting in

If I see "magnetic resonance" I don't immediately make the association with NMR, but maybe that's just me

@Secret hope you still see Ramanujan ..

@DavidZ I don't much either, but there isn't a perfect choice absent an increase in the limit

Yeah, agreed. Well, initials do play a significant role in getting people to recognize words, so I'd go for a three-part main tag name with the proper initials of NMR - I think it's safer to leave letters off the end of one of the parts. Given that we can't get the full name in, I don't know that it makes much of a difference precisely which part we leave letters out of.

@DavidZ but if we're cutting stuff out, why not just go directly for the initials?

they're a bit diminished in lowercase but they still pack most of the associative punch

For just nmr you mean? Because if someone who doesn't know what NMR is looks at that tag, they're completely clueless about what the tag actually means. Looking at nuclear-magnetic-resonanc or the like gives some idea of what it means.

@Secret you said “Ramanujan is particularly worse (he is much better now)” You mad bro 😠

is there any "real-life" homogeneous Lagrangian?

Maxwell perhaps?

hmmm

this looks like Euler's theorem

Ah, I finally got what the common past is

It's not just points that are to the past of every point, the points each have to individually be to the past of every single point

Hence $\downarrow I^+(p) = I^-(p)$, if the spacetime is causally continuous

Or $\downarrow I^+(p) \subset I^-(p)$ in general

@Fawad Well, since around 2016, I have not seen much help vampires other than koolman, so you are fine

lol