@BalarkaSen I spent like 3 hours stuck on a proof because I apparenly forgot what an inf does. I need to become an algebraist.

@JohnRennie how do I write $\mathfrak S$ on a whiteboard?

eh that's not even a bad one

Copy it from your phone?

I can't actually draw that

my hand can't move like that

carry a stencil with you

Don't look at me. I have trouble even writing block capitals legibly! :-)

Print it out, then cut out the character and stick it on the whiteboard with blu-tak :-)

i think if you really wanted it to look pretty a stencil really would be the approach. buuuuut

@Semiclassical Ok, consider a probability space. (Really a smooth manifold with unit volume.)

is that intended for here or the math chat?

I need a sequence of functions $\psi_i$ such that $||\psi_i||_N=1$, where $N=2n/(n-2)$ and $||\psi_i||_2\to 0$

@Semiclassical Here

$n\ge 3$ so $N>2$

I want the $L^N$ norm to be $1$ for all of them but the $L^2$ norm should decay

that's kinda weird

I don't know if it's possible...

the choice of $N=2n/(n-2)$ is pretty specific

@Semiclassical Blame Sobolev

ugh

I think what matters is that $N>2$

the only intelligent thing I notice is that $1/N +1/n = 2$

you mean $1/2$

yeah, you're right

That's the condition for Sobolev embeddings to work

not sure it really helps anyhow

ah

so it's motivated

Oh yes.

That relationship is fixed by scale invariance.

neat.

main reason I wanted to reorganize it like that was because it reminded me of Holder's inequality

but that's 1/p+1/q=1

sooo

Sobolev inequalities are just crazy Holder inequalities

yeesh.

functions on what domain?

In any case, I don't know if this is even possible

@Semiclassical A closed manifold

oh, right.

normalized so it's a probability space

so a simple-minded example would be [0,1]

Yeah that's what I want to consider first

yeah

Then rotate that and stick it in a coordinate patch

I mean the key is somehow that we want $$\int \psi_i^2\to 0$$ but $$\int \psi_i^N=1$$

to turn this around a little, if you have such an example then you can rescale the $\psi_i's$ so that the 2-norm remains finite but the N-norm diverges.

What's setting $n$ here?

@Semiclassical $n$ is the dimension

@Semiclassical Hm!

hrm. in that case the unit interval seems like a bad example, since n=1<3

@Semiclassical I don't think the dimension matters. Or the particular value of $N$. One just has to show what you said: If a sequence in $L^2$ is bounded it doesn't have to be for $L^p$, $p>2$

ah

so it'd be more general

I think so.

and what you said fits with what I remember about power means, with the AM-GM inequality being the obvious example

I'll confess, the main reason I want to take the 2-norm to be finite is so that I can set it to 1 and pretend I'm doing quantum mechanics :P

It's a dual problem. If you rescale by the $L^2$ norms then the $L^2$ norms are 1 and the $L^N$ norms should blow up.

sure.

I should definitely know how to do this...

hmm

if i indeed put on my QM hat and pick $\psi_n = \sqrt{2}\sin (n\pi x)$, then this has 2-norm of 1

Actually, it should be enough to show that $L^2$ is not continuously embedded in $L^N$.

But it's a larger space so that's nonsense

my thought above seems to not really lead anywhere useful

shrug

actually, here's a dumb example that seems to work: $\psi_i = i^{1/2}\chi_{[0,1/i]}$

that has $\|\psi_i\|^p = i^{p/2-1}$, if I'm looking at it right

oh I'm retarded

I almost had that

+1

woo

well now how to do that on a manifold. Oh well, I don't think this approach works for what I want anyway

trying to track down the original paper

Elsevier says their website doesn't work on Chrome. How rude

ugh

J. Math. Pures et appl.

what I find interesting about that example is that $\psi_i$ reminds me of a nascent delta function

I think one could make it work for [0,1]^n pretty well. but I can't imagine that's general enough

of COURSE this journal hasn't been digitized

why make life easy

i've run into that a few times

it's a pain

once or twice i've needed to get old math papers which were initially written in Russian and then translated to English

which usually meant I needed to head to our math library and dig through the bound periodicals

our library doesn't have physical copies of this one

of course not. that'd be too easy :/

@Semiclassical I've got this Sobolev inequality with a "best constant" but it's not straightforward.

I don't know how to turn it into a useful definition

I think once I grok what it really means the result will be easy

Namely, $\forall \epsilon>0\exists B_\epsilon>0$ such that for all $\psi\in H^1$, $$||\psi||_N\le (t+\epsilon)||\nabla \psi||_2+B_\epsilon ||\psi||_2$$and $t$ is the "smallest constant" for which this is possible

you can't take $\lim B_\epsilon$ either

it's a crap result

:S

I don't know how to characterize $t$

"the time when some weird-ass math happens"

@Semiclassical I'm thinking that there's an error

It's a little suspicious that no one seems to replicate this particular proof.

They all take very roundabout paths -- which I understand.

@Abcd no, I don't post on the physics forums site. I find it too full of junk compared to the Stack Exchange. Some other John :-)

Okay.

@Blue you're confusing me with someone who remembers anything about stat mech :-)

@MathematicsAminPhysics you don't think that asking nicely would be good? Maybe a few pleases and thank yous. After all the mods here didn't freeze/delete your rooms so why should they go to the trouble of unfreezing them just because you demand it?

Check out who came in the mail

@MathematicsAminPhysics No. All that happened there was you keeping them alive with useless posts consisting of single dots, anyway.

... and, moreover, please don't write in all-caps.

@EmilioPisanty Blue already said that ;P

well, I'll say it again, just for emphasis

rather

just to emphasize how one can emphasize things without shouting

=P

@ACuriousMind you know that diagonalisation question I asked

After a few hours I got back to thinking about it and realised there were still holes in my understanding of what you told me, and figured it's probably because chat tends to make things brief, and also that if I couldn't find the question maybe I should put it on the site, what's the stance on that? Putting questions on here on the site I mean

eh I posted it

if that stuff is frowned upon lemme know and I'll delete it

What's defined as a Physics workshop?

@Phase oh, please let's not get into that discussion again

@Phase where you work on Physics sitting at a shop.

@EmilioPisanty sorry if it's a heated / contested topic, I legitimately don't know

wait

That was me

What the hell

Ok well, referring back to what was said there

I have to come to the conclusion that a chat room would be a dreadful idea wouldn't it? Wouldn't it just be a chatroom full of multiple distinct conversations?

granted all about the same sort of topic but still

@MathematicsAminPhysics that means that the room you've just set up is inadequate for the purpose, as there isn't an actual meeting.

Is the Reissner-Nordstrom metric unique?

Is it the only spherically symmetric electrovacuum metric in GR

Please supply your own, explicit definition, and stop riding on the perceived benefits that arise from other definitions.

We've had this conversation before, multiple times.

Nobody is stopping you from carrying out your project, but it is your responsibility to make it work.

Spamming this chatroom with the same message is not an appropriate way to go about it.

@Phase I don't think that question is phrased particularly constructively. You've had multiple people give statements of the method, and you've found them confusing and/or not informative enough. What you're doing is yet again soliciting the statement of the method without giving any input about what you found confusing on the previous round.

I figured I'd just prompt in the comments for expansions on what I don't understand about the answer

@Phase I think that is an incredibly rude way to conduct the conversation.

oh

You're expecting people to churn out work for you that you already know you're going to find confusing.

it's not out of laziness, it's just because I don't really know what I'm getting confused on

Well, I don't

If you want a back-and-forth conversation where you can ask questions on the fly, bring it to chat

questions on main are not for back-and-forth

Ok, I'll see if I can delete the question

@Phase you can't

rip

improve it

explain how it's been presented to you before and why you found it confusing

I can't

maybe it's just my unique brand of idiocy but I find it difficult to exactly state what it is about this that leaves me confused

As it is, I might consider posting an answer, but frankly, the risk that you'll just go "yeah, I know you've already explained all of this but I don't really have enough concrete questions to help improve the answer, and yet I still don't find it helpful" is too big

That's not really it

I guess the only thing I can really communicate is that the request for some worked example, fully done is my statement of that

Because like Qmechanics answer

I could say I 'already knew that' but I just don't understand what to do with it

@Phase that's a much tougher ask than just an explanation of the method

I hope you realize that

How to actually think of that in a meaningful way

Well maybe my original phrasing example of the method is better

I dont really care about the specific matrix used just

clear, concise steps

I guess like one answerer put in his answer, I'm in the process of reading that now

Designing an example that's containable enough to be easily solvable but still has the just-right amount of nontriviality is not an easy task.

@Phase ::sigh::. It's like that 45-minute conversation two days ago never happened.

I understand that's annoying, and if you disagree that much flag and downvote my question, as it is I'm already annoyed enough at myself at having to reask this question alone. I don't really appreciate the feedback you're giving me, please just give it via site functions

If it gets closed I won't argue as I knew it probably wasn't a good question at the start

@Phase You were actively soliciting help and people spent their time specifically to help you personally, help which you're now disregarding entirely. I'm sorry if those people's annoyance at your disregard upsets you.

How am I disregarding that

Christ, it happened and it helped, I just wasn't able to glean a full understanding from that

Aren't you getting a little too offended on someone else's behalf?

I was explaining why I think your question is unconstructive and a bad use of the time of the people whom you're asking for help; you can take that feedback or leave it as you wish.

I mention that conversation again because you "don't appreciate the feedback", which is frankly a rather rude way to go about things. Again, feedback for you to take or leave at your discretion.

Nearly every question I ask is a "bad use of the time of the people whom [I'm] asking for help", just by virtue of it being poorly worded or a terrible question. In this case however it's because I'm just not sure what I'm not understanding, but when I try thinking through cases in my head I always end up not sure how to approach it. That's why I asked, again, for the FORMAL method, so that I could have something solid to commit to memory that went through the entire set of setps

*steps

I don't appreciate it because you're assuming lots of things

like a) that I even understand what it is that I don't understand

that's why i asked for a fresh new take on it, so that I'd have another chance of getting it, preferably with the full set of instructions

I don't see what's so criminal about my request.

@MathematicsAminPhysics Probably--link?

@Phase It's not criminal. I just think it is rude to actively hide relevant background from potential answerers, regardless of how unformed and unformatted that background is.

I don't have a background to provide beyond what I originally wrote

Like I said I don't have an understanding of what I didn't understand

Just that I don't HAVE an understanding of how to do it

@Phase A link to the relevant conversations in chat would make a huge difference.

As I see it that would just lead to getting potentially a worse answer

Because perhaps it's just the specific way @ACM goes about it that goes over my head. For all I know I'm just incapable of truly understanding it unless it's boiled down into "easy maths for dumbasses"

i wanted a new, full description

@MathematicsAminPhysics done.

@Phase See, that is a crucial piece of feedback for potential answerers.

But like I said, I don't know if that's what it actually is or whether it's any other things

> "The method as has been explained in (link) goes way over my head"

^ extremely helpful in knowing how to coach an answer

But I'm not sure if that's actually the case..

holy moly

But again, this is just feedback for you to take or leave at your discretion.

If you can't bear the fact that other people think you're going about things in an incorrect fashion, then I would suggest you stop caring so much about whether other people disagree with you.

I'm happy to drop this conversation at any time.

I don't understand where you're pulling that from, I just disagree with your feedback

I'm not aware of the reason why I as of that point still didn't understand the topic

whatevs

I hope you're lucky enough to find someone with the patience to explain this to you in a way that you will understand given the way you're going about this.

However I actually now DO have an idea of what I don't understand about something

Emilio I don't understand what it is that forbids me from just asking the question on the main site fully

Even if I understood it fully, but felt it was absent from the site

I wanted a new explanation so I asked for one. That's what it boils down to.

Hey, anyone want to answer an astronomy question of mine?

@Phase bejeesus. Nothing forbids you from asking it or from asking it like this. Other people disagree with how you're asking things and think it's rude. Deal with it.

Do SMBHs even contribute much to the kinematics of a galaxy? As in, if we removed one from the center of a galaxy, would the galaxy be noticeably distorted?

>forms opinion on something being rude

>forms opinion disagreeing

and I didn't write "ask" I wrote "just ask" i.e. without a qualifier of me trying to explain what I don't understand

I've given plenty of feedback already about why I personally think it's rude. I don't think further discussion would be constructive.

Sure, next time I'll put in information I don't have. ty

@SirCumference It's... been a long while, but I have a vague recollection (so may be extremely wrong) that this is how they were discovered in the first place

@EmilioPisanty This message is wrong by the way, and I responded as such but I feel you didn't see that. If you still care about this [which I doubt], I didn't ask for things I knew I'd be confused about, I asked for new explanations BECAUSE the previous one confused me. I would then comment if there were still things I didn't understand.

@SirCumference or rather, the orbits would be different, which I suppose isn't the same thing as the 'galaxy being distorted'. Again, best not to take my word on this :P

@Phase No ─ you didn't ask for new explanations, you just asked for explanations again (without taking any steps to guide the potential answers away from the explanations you'd already been given, which has already resulted in answers you found unhelpful). There's a crucial difference that you appear to be unable to see at this time, so I'm not sure it is constructive to repeatedly re-explain it.

I meant new in the literal sense, asking for another explanation, it's not as if one I already had wouldn't satisfy the question, and for all I know the same explanation explained ever so slightly differently could have worked.

I understand that generally you should provide as much information as possible to help people understand what it is that you dont, but I'm trying to explain that I didn't and still don't have that information

@Phase As I said above, the only information you really needed to include was links to the previous conversations and the statement that they went over your head.

You should have posted a link to the explanations ACM and Emilio provided you in the chat, IMO. "I have been explained here and here, but I wanted a formal, better explanation because I am not sure I understand it"

I presume that that's information you do have?

There's literally no effort required in doing that

^ that.

I don't think this is an unreasonable ask. But again, feedback for you to take or leave at your discretion.

I just feared that might skew answers more from what I wanted

@Phase From what I can see, the only thing what did do accomplished was to skew answers towards what you didn't want.

Why?

I got an answer that I like and has helped

^ Is this the kind of answer you were looking for?

Well no but I put in the question that I would preferably want an explanation of the method

because it is a direct answer to the question you did ask

> how does one define a formal way of going about simultaneously diagonalising both of them?

PREFERABLY WITH AN EXAMPLE OF THE METHOD

tbh you're just sounding like all the homework crew on the site who asks ten million people the same question and finally posts a question on the site without acknowledging any of the people who tried to help

@Phase If you mean Luc's answer, it is incorrect, btw.

oh god damn it

What's incorrect about it?

is it not general?

@Phase OK, so here's the thing. You're asking me to get into a back-and-forth over technical details of this method. The previous time you did this (i) you did not find it helpful, and (ii) you eventually got (from where I sit) extremely ungrateful about it. Why should I do this again?

i) who said I didn't? You told me what Block Diagonalisation meant as a term

ii) getting agitated over something unrelated imo isn't "ungrateful"

iii) if you deem i) and ii) to be true then don't, I'm not going to hold it against nor expect that if I did, it would bother you

That answer seems to diagonalize $A$ (to $A'$ say) by the usual basechange matrix $P$, then diagonalize $B' = PBP^{-1}$. But that's not what simultaneous diagonalization is... after I diagonalize $B'$ by basechange matrix $Q$, $A'$ becomes $QA'Q^{-1}$ in the new basis, which may no longer be diagonal.

i) Oh I see, I'm dreadfully sorry but I misattributed the conversation, from the original conversation we had about diagonalisation you helped me get more of a sense but by the end it just wasn't complete

and I'm not sure why

Bit stuck on this weird question... Says that if $\alpha (t)$ is a parameterized curve which doesn't pass through the origin. And that if $\alpha (t_0)$ is the point closest to the origin show that $\alpha (t_0)$ and $\alpha '(t_0)$ are orthogonal.

Pretty sure that this means that $|\alpha '(t_0)|$ is a minimum but I'm really not sure where to go from there?

@CooperCape Are you reading do Carmo?

yeah thou

trying to be less rubbish

@CooperCape Close, but not quite.

any suggestions...?

I am asking you to fix something you wrote in that sentence :)

Once you do it, what you wrote will be correct.

ummm

I'm flailing

Think for more than 2 seconds.

That $|\alpha '(t_0)|$ = 0?

No.

What does "$\alpha(t_0)$ is the closest point to the origin" mean?

$\alpha '(t_0) \neq 0$ ?

No, why?

it says that in the question as well

I'm really confused sorry

I'm literally just asking you what a part of the question means, aka, if you can write it down mathematically

Oh right oopsie

it means that the distance is a minimum right?

Im trying to teach someone with virtually no education in maths

I'm not entirely sure where I should start, but at the minute im going over the basic definition of a function, with examples like 2x etc

@CooperCape Yes, exactly. Can you write that down mathematically?

And when they're confident with that, introduce calc from first principles

That is, write down the function and write down where it is minimum.

am i being a dunce by skipping more basic stuff like fractions etc? Or can that wait

what's more important?

so maybe $\frac{d}{dt} |\alpha (t_0)|=0$

@CooperCape You're getting ahead of yourself.

What you want to say is $f(t) = \|\alpha(t)\|$ attains minimum at $t = t_0$

Okay

So we differentiate $f(t)$ and see that they are orthogonal

Now, as everyone does on first try, you're trying to invoke the minimization principles from calculus

right

@CooperCape No, here's the subtle point

$f$ is not in general differentiable

Think about $|x|$

right

Is it because of the way it flips around at x= 0?

@CooperCape So this, unfortunately, makes no sense.

Can you find a way to go around that?

There's a modification you need to do

@BalarkaSen Are you just being pedantic?

It's very clear what he means

No, I'm calling a nonsense expression nonsense.

I know what he means

Gimme like many a min whilst I try and figure out what's going on...

OK

Nope don't know what this modification is

Consider $\|\alpha(t)\|^2$. That's differentiable. Why? and what is the derivative?