@BernardoMeurer wtf

@0celouvsky What?

@BernardoMeurer Electronics engineering?

Really?

They're people too

I think

@0celouvsky They're closer in the family tree to me than electric engineers

You still betrayed me

How was I supposed to know?

I have this book next to bed and I'm proud of it

I have all five volumes of Spivak next to my bed and I'm proud of it

@BernardoMeurer you should have texted me

@JaimeGallego It's a nice book

@0celouvsky When your girlfriend asks me what EE means I answer

I don't want rebecca angry at me

I have enough women who dislike me

I hate HNQ

@ACuriousMind Be Nice.

Why do you hate it?

Because it rarely manages to pick questions I'd consider actually good :P

@ACuriousMind How would one show that $\text{Gram-Schmidt}:\mathscr B^\times\to \mathscr U$ is a homotopy equivalence?

$\text{Gram-Schmidt}$ is a pretty long function name

Unscr your math. What is $\mathscr{U}$ and what's the Gram-Schmidt function do?

Unscr?

I tried to say "don't use mathscr" in a slightly funny way :P

@ACuriousMind Why wouldn't I use mathscr?

@ACuriousMind The unitary group on a Hilbert space $H$ (that dependence is suppressed for notational convenience)

I don't like it. It looks...pretentious, like it thinks its letters are better than the others.

I happen to like using it for functional analysis.

Ah, well, functional analysis is better.

But my question stands regardless of my font.

*$\mathscr H$

What's the * for?

It indicates a correction

Oh

Yes, Hilbert spaces should also be $\mathscr H$

I was just too lazy there

$\vert - \vert$

@ACuriousMind Assume $\mathscr H$ is separable.

I still don't know what the Gram-Schmidt function does

@ACuriousMind nLab says that $\mathrm U( H)$ is homtopy equivalent to $\mathrm{GL}( H)$ via the GS process.

No further details are given.

Ah! I think I know what the map is - view a matrix in $\mathrm{GL}(H)$ as a choice of basis, then apply Gram-Schmidt to that basis, that produces an orthonormal basis which gives a unitary matrix when you write them as the columns of a matrix

Ah, ok

This is allegedly a defo retr.

Ah, yes, you can generally write a matrix as the product of an upper triangular matrix and the orthogonal/unitary matrix Gram-Schmidt produces

So the homotopy is just scaling the upper triangular part

(it helps to look at nLab's page for the Gram-Schmidt process)

Um, how does that work in infinite dimensions?

> 5. Categorified Gram–Schmidt process

Jesus christ

lol, maybe don't look at that :D

@0celouvsky What would fail in infinite dimensions (except the process taking infinite time, of course :P)?

@ACuriousMind Well I clearly can't have the process taking infinite time!

It has to happen in one minute

@ACuriousMind where are you seeing this?

@0celouvsky Seeing what?

Ah, that the homotopy is scaling down the upper triangular part? Just observe that the triangular part is topologically $\mathbb{R}^k$ (the triangular matrices are just a vector space), so you can just contract them

@ACuriousMind No, not that

the triangular bit

their proof that uses category theory (ffs)

@0celouvsky What, that Gram-Schmidt produces not only the orthogonal but an upper triangular matrix? That's called QR decomposition

Welcome to the wonderful world of linear algebra :)

I know about QR

I didn't know it worked in infinite dimensions.

Well, you can just multiply the bounded operator with the inverse of the unitary Gram-Schmidt gives to get the analogue of the triangular matrx

And these form a vector space so you can just contract that part and are left with the unitaries

@ACuriousMind idk this black magic

@ACuriousMind Is mathcal also pretentious?

No, it's stylish

@0celouvsky Maybe you need to offer your soul to some demon lord at this point

Uh, I lost that when I tried understanding Frechet spaces.

I'm supposed to "visually retract" an infinite dimensional set onto a simplical complex

"Since intuition - particularly for infinite-dimensional spaces - can be deceiving, we will write out this argument precisely:"

@ACuriousMind I recommend this book.

@JohnRennie Surprise paycheck!

So

For a circle

What is best

$S$ or $S^1$

(or $U(1)$ or $T$ or $T^1$)

$\mathfrak T$, clearly.

Are point interactions for point particles well defined

Like if I say that two point particles collide and just fuck off at a 90° angle

Is that coordinate independant

>index bundles

oh god

what

Are you doing vector fields of indexes

$A^{\mu(x)}$

it probably has to do with indices of Fredholm operators

but I wouldn't rule that crap out

"there is no epimorphism of $\Bbb Z$ onto $\Bbb Z\oplus\Bbb Z$"

what?

This may be more quantum related, but I haven't been able to find a list of all of the current solutions anywhere. I know a few of the implications of the solutions, but I wanted to know if there is a compiled list of them. Sorry to change the topic from Frechet spaces

a surjective homomorphism

Wonder why not

Oh, it would miss things like $(1,2)$

what do you mean by "a list of solutions"

Well, there are some looser ones like negative mass, then there's thinks like black holes we knew existed before finding physical evidence. Basically is there somewhere where I could find some of the solutions

*things

Are you asking for a list of solutions to Einstein's equations?

But then why saying it's quantum related

Sorry, should have specified, some of the implications are quantum physics related. And, yes I am asking for current, verified solutions.

of what, the einstein field equation?

There is a book, yes

"Exact solutions of the Einstein field equations", by Stephani

Yes, and thank you. I should've looked for a book earlier.

Although it's not exhaustive, but it contains a far amount of solutions

I suppose an article or even 20 wouldn't really be able to fit the solutions and math

Stephani is about 800 pages

exactly. Thank you.

it's good bedtime reading

also toilet

I'm just double checking here, is it:

amazon.com/…

yes

Thanks. I'll have to read it now. Bye.

not sure I like this

"Using symbolic manipulation software it is easy to verify that this manifold is Ricci flat"

Don't do it by hand, idiots!

Hm

Visser wrote a book on the Kerr spacetime

Should I get it

amazon.com/Kerr-Spacetime-Rotating-General-Relativity/dp/…

not the best cover

I never learned functional derivatives rigorously. Is this answer valid? physics.stackexchange.com/a/323733/148599

used copy for 30 bucks

Good enough

found a copy at 17€ including shipping

noice

I should get Chandasekar's book too someday

@PPenguin No, your comment is correct.

ok, thanks

The rigorous definition of a functional derivative is $\lim_{\epsilon\to 0}\partial_\epsilon f(x,\epsilon)$

What Qmechanic said is right

For instance in GR we have annoying boundary terms that come from the derivative of the metric

Hm

BTZ black holes have CTCs in them

And I think Carlip did some quantization of the BTZ black hole

I wonder if that feature stays in it

@0celouvsky Qmechanic says the boundary conditions must be set appropriately, although I have seen other arguments that claim the fields must die off fast enough (physics.stackexchange.com/questions/57313/… ).

@PPenguin Yeah so that's because physicists like to do their integral over the whole space

In more mathematical contexts you can avoid it by doing action principles locally (i.e. on precompact open sets)

@Slereah so the construction of $\Bbb Z$ from $\Bbb N$ can be paralleled to create a group of vector bundles over a space

@0celouvsky Thanks for helping.

wot

@PPenguin and honestly I don't see a good reason why arbitrary fields should die at infinity

arxiv.org/pdf/hep-th/0506104.pdf

not what I wanted but could be nice

etheses.whiterose.ac.uk/1646/1/Leonardo_Ortiz_PhD_Thesis.pdf

If the universe is infinite and homogeneous, why should some field be very small in one place but not another

So the integrals converge?

Please, mother nature doesn't care about our integrals

Plenty of fields don't die at infinity

$\phi^4$ even admits solutions that are just constant everywhere

(and non-zero)

@Slereah proof?

@0celouvsky you can easily check that the Jacobi elliptic function is a solution

For which there are solutions that depend only on t

@Slereah are you serious

The jacobi elliptic function is literally a solution to $${\frac {\mathrm {d} ^{2}y}{\mathrm {d} x^{2}}}+(1+k^{2})y-2k^{2}y^{3}=0$$

It's not rocket science

idk, is that the $\phi^4$ equation?

$\phi^4$ is $$\Box \phi = g \phi^3$$

@Slereah So the index bundle is a way of assigning the index of a family of Fredholm operators parametrized $X$ to a vector bundle on $X$

@Slereah that doesn't look like what you wrote above?

arxiv.org/abs/0907.4053

if u want proof

@Slereah So the index bundle

it's pretty shit

@Slereah what's on page 54 of Steenrod?

@ACuriousMind @BalarkaSen So the fact that $[X,\mathscr B^\times]=0$ implies that any vector bundle over a compact manifold with infinite dimensional Hilbert space fibers must be trivial.

@0celouvsky a photo of a turtle

I doubt that.

lemme see

either something about reducible spaces or the construction of a cross section

hmm

oh well

I think I need that if the structure group is contractible, then the bundle is trivial

or maybe that homotopy classes of maps from the base to the structure group do something

@Slereah what is a jet bundle even supposed to do?

itisamystery.com

$\mathrm{Hom}(V,H)\cong V'\otimes H$?

@ACuriousMind Does that hold for $H$ infinite dimensional?

I'm sure it does, because $L\in\mathrm{Hom}$ is defined by finitely many values on basis vectors of $V$.

Risky topic, mister Gourgoulhon

You're reading that book now?

Good luck

It's a brick

well, not all of it

I don't know why I bought it

For some reason a lot of paper on the Alcubierre metric revel in using little spaceships in examples

they can't just calculate geodesics

it has to be a little ship going to the stars

because they're all insane.

@Slereah Fubini's first name is Guido

v. Italian

why the hell do Fourier series even work

I never looked up a proof of 'em

You need to show that if $\int_0^1 e^{i2\pi n x}f(x)\,dx=0$ for every $n$, then $f=0$.

After that it's basic Hilbert space theory

oh, $f\in L^2[0,1]$

Ok, the proof is in Conway's functional analysis book

It uses the Stone-Weierstrass theorem + density of $C_0^\infty$ in $L^2$

and on that note I have to go to sleep

cheerio

@Yashas: snap! :-)

o0

processing the reply; still trying to decode what that reply meant @JohnRennie

someone flagged one of my chat replies? o0

@Yashas Ah, oops. Snap is an English card game - you shout Snap! if the two players draw the same card. It's used colloquially to mean two people have just done the same thing. I said it because we've jut both posted the same comment to that question about eye movement.

oh :P

@0celouvsky So are you going to lash out $200+ on that book?

@0celouvsky What's $B^\times$

The monoid of Breal numbers under multiplication

If you consider the spacetime manifold of classical mechanics, does the projection of the spacetime on the slicing have a name

The projection $(t, \vec x) \to \vec x$

Hi :-)

Morning everyone.

Apologies if my responses are a bit slow at the moment. It's a little lively at the office!

@SwapnilDas Morning :-)

Hey :)

Very cool.

How about your JEE?

My distant senior friends have scored 300, 323, 320 :P

Super humans.

I'm happy to have joined the JEE world now.

Lol.

I've ordered Cengage, I hope it's a good one.

Hmm.

The philosophy StackExchange chats are fairly empty

@blue I have finished studying 50 pages out of 500 :D

1.5 days left for the exam

also the history of science one

Physics is addictive :(

well, economics questions are stupid

"If total revenue at 2 units of output is Rs 200, what is average revenue?"

"6. How is the midpoint of a class interval obtained?"

You can answer this one

"Distinguish between consumer goods and producer goods. Give examples."

The bad questions are 8 mark questions

Explain average value

wth?

what the hell do I write for 8 marks?

My sister in law is an economist, well an econometrician. It's a well paid job. She earns a lot more than I ever did :-)

There are some nice 8 mark questions too

"Describe the effects of population growth on natural resources of a country."

I can write whatever I want

@blue I remember writing Newton's laws of motion in my social science exam in 9th grade :D First point and the last point are relevant to the question, the rest are physics.

I did that 3 times

got caught the 3rd time

and I was surprised to find out that another 5-6 people got caught with me lol

out of a class of 40

that's like 1/5 of the class

xD

@blue I must admit that a career as an econometrician has never interested me.

@blue Maths. I almost did a maths degree - I was offreed a place to do maths at Imperial College in London. But at the last minute decided to do science instead.

@JohnRennie why did you decide not do math?

Economics is p. boring but on the other hand

All that money

Although that might be a meme

the same way that getting an easy job with software engineering was a meme

@Shing I applied to Cambridge and Imperial College to do maths but Cambridge didn't make me an offer. But at the last minute Cambridge offered me a place to do Natural Sciences instead of Maths. I really wanted to go to Cambridge, so I took their offer.

Apparently there is another break of causality in classical mechanics, similar to Norton's dome, called the "space invaders"

@Shing It turned out a really good thing because the school Maths lessons hadn't prepared me for what a Maths degree would be like and I would have hated it. So everything turned out for the best even though i was disappointed at the time.

Ah, apparently the space invader is like

1) take an object accelerating from rest

2) have a totally unbounded acceleration

3) reverse the time axis

the unbounded acceleration object has the trajectory

while the space invader has this one

The object comes out of nowhere at $t^*$

@JohnRennie Sounds like what IITians do

but it is far worse

neat

People who are interested in Computer Science take pulp engineering just becaz they want the IIT tag lol

biology? omg

I wouldn't take biology even if I was offered one at Harvard or MIT

@Yashas To be fair, part of the reason I wanted to go to Cambridge was for the prestige. And it works. When I was applying for jobs the recruiters tended to see the word Cambridge and immediately offer me a job.

101% Guarantee that I would fail

@JohnRennie The same happens with IITs too; good companies rush to the gates of IITs. The are also engineers who graduate from IITs become politicians, writers, etc.

They take the IIT tag and then do what they really wanted to do.

@Kaumudi.H You can change your branch if and only if you have high CGPA

And there are more rules; for example, IIT Bombay has a rule that the size of a class may not increase or shrink by more than 10%.

You don't get that choice until your second year.

@Yashas the main thing about Cambridge is that it's so amazingly enjoyable. They work you really, really hard - basically 9-5 every day and Saturday mornings - but it's so rewarding, and you can talk to Nobel prize winners about your course work :-)

@JohnRennie IITs are enjoyable becaz there is no rule for attendance. If you score good with 0% attendence, you are going to graduate :D

One of my tutors was Aaron Klug, who won the Nobel prize for Chemistry :-)

:O

:O

Amazingly the Nobel prize winners are (mostly) regular guys and (mostly) very nice people.

I want to meet Feynman >.<

Get a spade

lol

That joke copyright © Bad Taste Jokes Inc. :-)

actually I am not quite sure about my future. Due to heath issue, I have just graduated with a Physics bachelor at a age 27

I prefer research

but it seems a bit too late to do a PhD at my age

by the time I doing a Postdoc, I will be around at least 34

or 33

and the overall academic environment doesn't seem good.

Apply for IIT :D

It is funny that NIT admits foreigners who have 2400 in SAT.

They can't appear for Mains.

There are innocent students who were born in another country; they prepare for the JEE for years and later find out few months before the exam that they are not eligible.

Hm

I'm not 100% sure the space invader scenario makes all the sense

It involves discontinuous forces

I mean I see where they want to go with it, but

Mathematically I think it might be poorly thought out

For those interested in a failed mathematician's life

Yeah, ending with suicide is not the cheeriest perspective.

@blue It didn't strike me as pessimistic. He obviously remembers his years at Princeton with great affection.

The closing paragraph is a joke. It's a variant of the joke about Hilbert.

How morbid.

Students committing suicide comes the territory, I guess.

I mean have you even read about the jet bundle

how do you not get depressed after that

Thanks to JEE.

Joint Extermination of Enjoyment

@blue I'm sure that number goes up once JEE sends out the results of their exam.

Double integral of the tangent is some awful stuff

@Shing All through school and undergrad you are being taught i.e. you sit there and someone tells you waht you need to learn. The big change in a graduate degree is that now you have to make the decisions. You have to plan your experiments and decide what to do.

That makes a graduate degree far, fasr more rewarding than doing your degree.

But of course it can be a bit scary. So I'd say the most important thing is a supervisor who gives you the freedom to plan your own work, and make your own mistakes, but keeps enough of an eye on you that you don't go ciompletely wrong and end up with nothing.

@ACuriousMind quick note, this may be useful in the future, particularly the part that puts some numbers on what SE officialdom deems to be an 'extreme number of downvotes'.

@JohnRennie agree... freedom to make mistakes is so important in studying.

Any tips on choosing a supervisor?

@Shing chat to their students!

^

The proof is in the pudding.

maintenance got over in 30 seconds? huh

@JohnRennie Would they mind receiving my email? Or worrying about telling me plainly ? (say, if they are in America, and I live in Asia. )

I just wrote, erased, and rewrote a u five times before I didn't accidentally write mu instead

Q_Q

I am planning to write to a prof, about a position of research assistant in his research team, not exactly sure the important part.

I have no experience in experimental physics...

You need be careful about anything I say because my experience is 40 years out of date, and I've never applied from a different country. But for what it's worth ...

If, and only if, you are offered a place or seem likely to be offered a place ask the supervisor if you can talk to a couple of his students to get an idea how things like accommodation work. If they agree then you can e-mail the students and see what they say.

But be careful what you ask. If you ask the students whether they think their supervisor is a son of a bitch they'll (a) say no and (b) tell the supervisor you asked!

Feynman wheeler's absorption theory isn't good to read when tired

Lots of text, not a lot of equations

"The sun would not radiate if it were alone in space and no other bodies could absorb its radiations... If for example I observed through my telescope yesterday evening that star which let us say is 100 light years away, then not only did I know that the light which it allowed to reach my eye was emitted 100 years ago, but also the star or individual atoms of it knew azlready 100 years ago that I, who then did not even exist, would view it yesterday evening at such and such time."

Time symmetric theories are slightly weird

Stay away from walls of text when you're tired.