Mathematics - 2017-09-07 (page 4 of 6)

@RE60K Right. So $\phi(u+v)=\phi(u)+\phi(v)$.

RE60K

Thank you!

np

it does seem strange to ask it this way, though. I'd have more expected them to ask you to show that it's an automorphism

whats an automorphism guys

I know hom and iso

domain = codomain

5:03 PM

Hmm as in G-->G

right

What is so special about that

Most of the example on homomorphism we got were auto then

:D

or wait ._. we had R* and R^+

yeah, that's a bit different.

wiki's got a page on automorphisms: en.wikipedia.org/wiki/Automorphism

@Kasmir: An isomorphism from $G$ to $G$ is often called an automorphism of $G$.

@TedShifrin Thanks :)

Mary Star

5:13 PM

Hello!!

We have the double integral $\int_{0}^{6} \left (\int_{\sin^{-1}{\frac{y}{6}}}^{\frac{\pi}{2}} f(x,y) \, dx \right )\,dy$ and I want to change the order of the integrals.

We have that $0\leq y\leq 6$ and $\sin^{-1}\frac{y}{6}\leq x\leq \frac{\pi}{2}$

We get $\sin^{-1}\frac{y}{6}\leq x \Rightarrow \frac{y}{6}\leq \sin (x) \Rightarrow y\leq 6\sin (x)$ and so $0\leq y\leq 6\sin (x)$ and $0\leq x\leq \frac{\pi}{2}$.

So, do we get the double integral $\int_{0}^{\frac{\pi}{2}} \left (\int_{0}^{6\sin (x)} f(x,y) \, dy \right )\,dx$ ?

I got few exercies on homomorphism and isomorphism I wish you could help me with Ted @TedShifrin

@TedShifrin By help I mean ofc, just clearifying things , not give me the anwer >< because that way I wont learn anything :D

You know I'm stubborn.

Especially now, cuz I just had a tooth removed and I'm bleeding :P

@TedShifrin Oh :( I hope you get better soon

Alessandro Codenotti

Hi @Ted, do you also prove Fubini in your multivariable book/videos?

Oh, that sounds painful

@Alessandro: Only for Riemann integrals and with strict hypotheses. Not what you would consider Fubini.

But good enough for Riemann stuff.

Perfect @MaryStar.

Alessandro Codenotti

5:16 PM

We did a more general version and the proof is very complicated and full of nasty details

You did Lebesgue integral, surely?

Yes, Fubini is technical ... including completing the product measure and stuff I've totally forgotten.

Mary Star

@TedShifrin Great! Thank you!! :-)

Alessandro Codenotti

Yeah, we did Fubini with the product of two $\sigma$-finite measures (but any actual example or exercises used the Lebesgue or Hausdorff measures)

Let G be a group . Prove that the following function is an homomorphism if and only if G is abelian, @TedShifrin What is the proof teknique here ? I assume first that it is an homomorphism then I show it is abelian ? then the other way around ?

Sup chat

5:18 PM

Yup. As always with if and only if proofs. I assume $\phi(g)=g^{-1}$. This is standard and not hard.

nm Eric wbu ?

heya Eric.

@TedShifrin Okay , the thing is that I dont have much expericnce on this kind of proofs=p

@TedShifrin the function is from G--> G , how to define phi ?

Just start with and use definitions.

Not long till school starts up again, cramming in some geometry cause I won't have much time to study for fun

5:20 PM

They gave you the homomorphism, right, @Kasmir?

Oups ><

Yes yes :D

I hope your family will be safe in Florida, Eric.

that part should not be done =p ok working on it now ! :D

Me too, I'm actually really on edge :(

My sister was actually down in Florida as of last week rehearsing for a show

5:21 PM

@TedShifrin Are you gonna be here long btw? its important so I can change my schedule for today =p

Yeah, I'm sure you are. But, don't worry. Rush Limbaugh says it's all fake and not to worry.

Our home can't handle the hurricane force winds so they're staying over at their boss's house

but once word on the hurricane came they cancelled it and had everyone leave town

Kasmir, I'll be around off and on today. Recovering from surgery so not doing much.

my sister was driving up to atlanta yesterday and was going to get a flight from there.

5:22 PM

Omg has "fake news" extended to include weather????

so she's out of the way now

Scientists don't know nothing, Eric. According to Trump, Rush, and all their cronies.

@Ted did you watch Federer lose to DelPo? Was the match any good or was Federer just weak?

@EricSilva they already had climate as fake news. jumping to weather isn't such a jump

Semiclassic: I read that the airlines were gouging people unbelievably to get out of FL.

5:22 PM

@TedShifrin Okay , I hope you get better soon ! ( Am saying this from the heart ) =p

details from WaPo here: washingtonpost.com/news/the-fix/wp/2017/09/06/…

My parents and Aunt are directly in the path of the storm and can't afford to get out so I've been trying not to freak out

@Danu: I missed the last set and a half. I have recorded it. Federer was making errors, and DelPo was hitting him off the court they way he did in 2009. It's hard not to be happy for DelPo. Amazing resolve.

I'm pretty happy that it's him that beat Federer; my second favorite player :)

I like him ... and he has learned to volley adeptly.

5:24 PM

and to clarify, Rush didn't say "there's not going to be a hurricane" but "no way it'll be as strong as they say." still stupid, but not quite as monumentally so

Yikes, Eric, I'm sorry.

@Semi wow that makes me furious, hurricanes are unbelievably scary and this suggestion is absurd, it could seriously hurt people

yeah

Semiclassic: It's all pretty monumental, if you ask me.

I mean, there is some truth in that the cable media looooves storm coverage

5:24 PM

When I was younger a hurricane destroyed my family home and we were left homeless for a while, and that was a category 3 significantly weaker than irma

but the idea that the magnitude of the event is overstated is just idiotic.

I mean, my New Years trip to ATL last year when they forecast a bad ice storm and I left days early and cancelled all sorts of plans ... and then nothing materialized. Weather forecasting isn't perfect, of course. But hurricanes of supreme strength ... gotta take

'em seriously.

right.

Damn, Eric, that's tough, man.

when is irma forecast to hit florida?

5:25 PM

This weekend

@Danu: You saw my Cartan formula reply to you and Balarka?

@TedShifrin What I found weird is that, while Arnold has Cartan's formula in his classical mechanics text, he refers to it as the "homotopy formula" and seemingly relegates it to a problem

What's this abt one of the coolest formulas in math?

Well, it is a chain homotopy from $\mathscr L_X$ to the $0$ map.

right.

and the hint he gives for that problem is to that effect

5:28 PM

@Ted We were discussing about Cartan when I mentioned that intepretation

See Balarka's starred remark to the right, Eric. But you and I have talked about this for first variation of arclength/area.

Ohhhh, OK, Balarka. I shouldn't jump in where I'm not needed.

Yeah we were discussing it, I"m aware it's Cartan's formula :P

But I didn't know it was a useful interpretation and that you use it quite as often :)

apropos of nothing: ugh, i forgot how annoying the signs can be when doing elementary schrodinger equation stuff

It's great for variational computations.

5:29 PM

Yeah those variational calculations are so slick with cartan

Signs and factors of $\sqrt{-1}$ are always annoying.

Glad I brainwashed you, Eric :P

^

Schrodinger equation has i's too :P

Even Chern messed up a sign with a $\sqrt{-1}$ in the first edition of his complex manifolds book. That was one of my main contributions in fixing it for the second edition :P

with regard to variational calculations, I have a sneaking suspicion that the right way for me to learn riemannian geometry would be to learn geometric optics :)

5:31 PM

Seriously, the last major problem in my thesis is determining a single sign

It's always confusing in complex geometry because we're so used to $1/(2\pi i)$ and sometimes it needs to be $i/2\pi$.

I say with only a little bit of sarcasm

I think my love of computation makes me rlly like the cartan game bc of how slick everything turns out to be but I value knowing a lot of ways to get the same result

with the Schrodinger equation it's annoying because the usual one is $i\hbar\,\partial_t \Psi = -\dfrac{\hbar^2}{2m}\partial_x^2 \Psi+V\Psi$

No love deep computations

5:32 PM

@Eric I hope your family will be safe

Thanks @Daminark

What's happened?

@semi do you know anything abt GR

but the calculation right now also needs the complex-conjugate of that, so $-i\hbar\,\partial_t \Psi^*=-\dfrac{\hbar^2}{2m}\partial_x^2 \Psi^*+V\Psi^*$

Do you happen to know whether the characteristic classes of (prominent) symmetric spaces are computed somewhere @TedShifrin?

5:33 PM

I know I don't know much :/

I just picked up Bob walds book and was interested to go through it properly

neat

I do not know, @Danu, but it seems like the answer should be yes.

@TedShifrin the function G-->G that sends x to x^-1 , so assuming f (xy) = f(x) f(y) = x^-1 y^-1 , but from here I dont know how to continue to show abelian, just a hint please

I've never had a GR course.

went down the quantum route

5:33 PM

There's a hurricane going to Florida, and it's really strong @Balarka

I've seen the diff geo material in it and raged when he said christoffel symbols were tensors

@Kasmir: What is $f(xy)$?

^

@Daminark Oh damn

you've written down $f(x)$ and $f(y)$ but not $f(xy)$.

5:34 PM

But I'm interested to know some of the physics applications of my favorite subject I guess

Yeah, Eric, we discussed that. I would toss the book overboard.

@Balarka it's very scary and my family is directly in the path of the hurricane basically

@TedShifrin x and y are two elements in G, took the product xy and mapped it in G

@Kasmir: Answer my question.

@EricSilva They're moving out of it's way, hopefully?

5:35 PM

I'm trying to determine the Pontryagin classes of $G_2/SO(4)$. The signature theorem tells me $1/45(7p_2-p_1^2)=\sigma(M)=1$ and the index theorem means that $\hat A$ vanishes, hence $7p_1^2-4p_2=0$. Since $\dim M=8$ and the cohomology is $\Bbb Z$ in degrees 4 and 8, this determines $p_2=7 g_8$ and $p_1=\pm 2g_4$ where $g_k$ is the "positive" generator in degree $k$.

@Eric take GR and get on Wald's case for it

@TedShifrin it is an homomorphism

@Balarka they can't afford to make it out unfortunately

The only thing that's left for me to determine is the sign on that $\pm 2g_4$...

Read the question carefully, Kasmir.

5:35 PM

So it's just waiting to see what will happen

@Daminark ive been suggested this numerous times

@EricSilva: You might check out the book on relativity two Berkeley guys wrote — one relativist and one geometer. Sachs and Wu.

@TedShifrin what is f(xy), it is the image of xy

@EricSilva re: geometric optics as an application of variational principles and Riemannian geometry, I'm actually rather serious

@TedShifrin (ノ゜Д゜)ノ︵ ┻━┻

for instance, I saw this book the other day: books.google.com/…

5:36 PM

@Eric yikes.

I hope they stay safe

@Semi actually I've been told that I might like it by my analysis professor whose a mathematical physicist

I am interested to learn more physics on the whole, I've taken one intro course in my life basically

Schlag?

@Balarka thanks me too :/

the basic point being that geometric optics can be founded on Fermat's principle of least time, and that's a variational principle

Yup @Daminark

5:37 PM

@TedShifrin I really don't know what you want me to answer , f is a function, x and y are elements in G, f(xy) = f (x) f(y ) because it is a homomorphism

What is $f$?

Variational things are one of my favorite parts of math @Semi so I'd be interested to check it out

you get way too many hurricanes on your part of the world

a function

@Ted I'll see if I can find that book!

5:38 PM

Fake weather, Balarka.

a map

WHAT map?

fun fact: I actually learned about the connection between geometric optics and the brachistochrone in high school

@Balarka I think there are 3 in the Atlantic basin right now

What does that function do to elements of $G$?

5:39 PM

Wait it's been a long time since I remember this many hurricanes

f :G-->G that sends x to its inverse @TedShifrin

@Eric man

@Kasmir: When you're learning to write proofs, make sure you use all the information you have been given.

So? What is $f(xy)$?

@Semi an analysis professor I TAed actually included a bunch of physics problems on the brachistochrone I thought were quite nice

yeah

5:40 PM

Esp cause analysis here is usually super abstract

@TedShifrin am not assuming abelian yet , gonna prove that part

You're not hearing us. @KasmirKhaan

I even mentioned the brachistochrone/tautochrone in my multivariable videos.

Kasmir. I'm not answering any more.

@Balarka I grew up with hurricanes being a constant fear come summer time, it really is scary

is brachistochrone that curve along which a particle slides down in the fastest possible time

5:40 PM

yea

By definition, $f$ sends $x$ to $x^{-1}$ and $y$ to $y^{-1}$. If we write $z=xy$, then what is $z$ sent to?

what is f(xy) is (xy)^-1

the origin of variational calculus!

what else you want me to say

And tauto- , Balarka?

5:41 PM

there you go

you hadn't said that yet

omg

was that it ?

OK, finally, Kasmir. You should have said that an hour ago.

:(

almost.

@Eric in this part of the world we fear heatwaves in the summer lol

5:41 PM

I thought you had something else in mind

And what is $(xy)^{-1}$?

did not expect such question :D

I told you it's just a matter of using definitions, didn't I?

y'x'

Yes but ><

@Danu gotcha

5:42 PM

OK, we're done.

Oh those are a problem in Florida too tbh, maybe not as much but I know a few people who have nearly died from the heat

my mind went to other things

@TedShifrin why ? :(

I don't want to hear excuses.

@Ted I don't think I know tautochrone

Sorry (

5:42 PM

he means "you're in a position to finish now"

am really trying my best

^^

Oh

If we had a calculus of variations course here I would take it above almost anything else tbh

@Eric a nontrivial proportion of the population die out of heatwaves in the summer here

5:43 PM

thought he was "done" helping me =p

Scary

Ok good let me finish my proof =p

you've just shown that $f(xy)=(xy)^{-1}=y^{-1}x^{-1}$...

@Balarka: Tautochrone ... takes the same time to get to the bottom no matter where it starts!

Do a reading course with Soug @Eric

5:43 PM

We at least have air conditioning in literally every building

@TedShifrin Ohhh

I think I have seen that at some point of time

@Daminark tbh I might

I actually built a model of a brachistochrone in high school

(translation: my dad helped me make one)

I was amazed when I was first told that ... It's actually in Simmons's beautiful differential equations book, if I remember correctly.

Yeah I wasn't really being sarcastic, like it'll be hell but you'll learn a good bit

5:44 PM

It was a cycloid track with a straight track attached to it

@Ted This is like the Huygen's pendulumn isn't it

pendulum

I have a nontrivial interest in analysis of nonlinear pde and souganidis could be a good dude to learn it from

if you put marbles on it, you could see that the marble on the straight path took longer than the cycloid

Oh, @Balarka, I'd forgotten that. Is it?

Huygens's pendulum is an exercise in my multivariable book — the evolute of a cycloid is another cycloid.

you could also put marbles on both sides of the cycloid track and see that both reach the bottom at the same time

regardless of the starting heights.

5:45 PM

@TedShifrin If I start with f ( (xy)^-1 ) to get xy, is that maybe better? then I get xy=yx instead of y'x' = x'y' , what do you think ?

@Daminark it'd probably be a fun time tbh, I think his philosophy of doing literally obscene amounts of problems isn't bad if it's one on one

Perfect, Kasmir.

:D

thanks my mind working better now, under stress =p

@TedShifrin Even though the amplitude dies out, the time period remains the same for all time. It's kind of similar, isn't it?

Or you substitute in the final $x^{-1}y^{-1} = y^{-1}x^{-1}$ or you take inverses again. Whatever.

5:46 PM

But do I have to like put that x'' = x ?

One reason I really should read up on geometric optics is that it does naturally connect with semiclassical approximations

or I am allowed to assume that?

$f(f(x))=?$ @KasmirKhaan

@Balarka: Remind me of Huygens's pendulum? I was thinking the pendulum swinging against the cycloid.

@Kasmir: You know that, just like you know $(xy)^{-1}$.

@Semiclassical f(f(x))= x in this case =p

5:47 PM

Yeah it's the exercise from your book. Put up a bob and swing it against a cycloidal frame

in every case, for this $f$.

@TedShifrin Okay, I want to learn how to make perfect proofs =p no flaws

The frequency/time period is independent of the amplitude

Okay I got it thanks :D

Ill try to do the rest now

OK, @Kasmir.

5:50 PM

ugh, so that's where my sign error was

How does one go about integrating something like 4x/(x^4-1). I'm not sure how to start?

What techniques do you know, @10Replies?

I'm about 2 weeks into calc iI

*II

That doesn't answer my question :)

U substitution, Kinda know integration by parts, barely know partial fraction decompesition

5:56 PM

You could certainly do partial fractions.

I would rather do a trigonometric substitution, but I don't know if you've done that and I don't know how it comes out.

I have done a little bit. I don't really understand trig substitution fully yet

I would try $x^2 = \sec\theta$ and see what happens, but this seems a hard problem if you've just started.

The way I see to do it is to first do a u-substitution (not a trig one) and then do partial fractions

but it seems hard for a two-weeks-in problem

It's in my calc homework :/

That's probably what they are expected to do, Semiclassic. I thought of that immediately, too.

SteamyRoot

5:59 PM

Can't you split in partial fractions?

@10Replies, do you know what Semiclassic is suggesting you try?

It's a mess, Steamy. Semiclassic turns it into $du/(u^2-1)$ quickly, and then partial fractions is easy.

SteamyRoot

Well, yeah, after that of course

Damnit, I got sniped >.<

@TedShifrin right. except for a factor of 2 :)