Mathematics - 2017-02-06 (page 5 of 5)

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11:00 PM

there seems to be no correlation lol.

Il faut qu'il s'en aille chez un psychiatriste, quoi.

$b/1$ goes to $a/c$

@Zach: Oh, but there is. And you will kick yourself ...

$a/1$ goes to $b/c$

ok so

$a$ and $b$ switch

$1$ and $c$ switch

Aha ...

11:01 PM

is that it?

Now how should you have seen in that in one millisecond?

i shouldn't have

But yes, you should.

Akiva Weinberger

Wait until he's got the final answer first…

wait

11:02 PM

He does.

Maybe write it down, though, Zach.

ted when I do that i get wrong answrr

ok so what about for quadratic

I did dz last

$x^2 - ax + b$

tripple integral of r dr dt dz

11:02 PM

Well, I'd do $d\theta$ on the very outside, @Kasmir. I'd do $dr\,dz\,d\theta$.

@Ted Oui, le problème c'est qu'on peut pas le faire à sa place

OK, that's fine. What are your limits.

Enfin bon, je vais arrêter avec mes histoires morbides

sum of reciprocals of cyclic sum of order 1 is just $1/p + 1/q$ which is $p+q/pq$ which is $a/b$, while normally its $a/1$

Ton ami a un peu de chance que tu sois un bon ami, @Astyx. Mais il faut un professionel quelconque.

@Zach, you had the switches correct. So write down your new polynomial, please.

11:04 PM

sum of reciprocals of cyclic sum of order 2 is $1/pq$ which is $1/b$, while normally it's $b/1$

oh for the cubic?

1<r<2 , 0<t<2pi , <z< , i cant figure out the boundries of z , i did them wrong first time

it turned to umm

let me see

@Kasmir. You're violating the basic rules. Start on the outside and work your way to the inside.

$cx^3 - bx + ax^2 -1$

?

Akiva Weinberger

Yes!

11:05 PM

oh wait

IT'S REVERSED?!

Uh huh. Now tell me how you should have seen that in a millisecond.

hmm

well in each one of them

it was divided by $pqr$

Forget how you derived it. Start over.

observations take hundreds of milliseconds

LOL, tern.

Yeah, it's turning into an hour-long millisecond.

11:07 PM

uhh

using vieta's formula or what?

@Zach: I asked you for the polynomial whose roots are the reciprocals of the roots of $f(x)$.

NO Vieta.

Akiva Weinberger

@arctictern How many lightnanoseconds away is this polynomial

poor viete

all he wanted was for people to use his formula

@AkivaWeinberger 1/c

what are we doing

11:08 PM

Nothing.

let me just

think

Akiva Weinberger

I always confuse Vieta and Viète

Viete is the guy who made Vieta's formulas!

Akiva Weinberger

(the latter is the guy with the infinite product for 2/pi)

Trying to disprove Riemann Zeta by manual inspection of zeroes

11:09 PM

it was?

wait

i kerfuffled myself real good.

Back to my question, dammit.

aughhhh

Akiva Weinberger

@MikeMiller Giving Zach a problem with roots and symmetric polynomials

I don't want one millisecond turning into two hours.

Try quadratics. If f(x)=(x-u)(x-v), what is the factorization of a polynomial with roots 1/u, 1/v? Can you write the factorization in terms of f(x)?

11:10 PM

tripple integral of r dr dt dz = 1/2 * 2pi , integral of pi dz @TedShifrin

I want limits on integrals, @Kasmir. Come on.

($x-1/u$)($x-1/v$)

Keep going, @Zach.

Akiva Weinberger

Your thing wasn't monic

viète and vieta aren't the same person ???

11:11 PM

yeah i thought they were

anyways ted will yell at me so i must continue

Akiva Weinberger

@mercio $2/\pi$ guy and symmetric polynomials guy

One sounds French and one sounds Italian. But one can rarely trust history.

$x^2 - x/u - x/v + 1/uv$

@TedShifrin 1<r<sqrt2 , 0<t<2pi , 0<z<1

Akiva Weinberger

Coka-Cola sounds simultaneously French and Italian

11:12 PM

@ZachHauk colder

The

According to wiki, same guy.

try linear functions first. how to write x-1/u in terms of x-u?

NOOO, @Kasmir. Come on. Fix z, fix $\theta$. What does $r$ do?

The Amin principle: If a theorem has a name attached to it, the named person isn't the one who invented it

Akiva Weinberger

11:12 PM

@TedShifrin Wait what

French guys, french guys everywhere

Note: The Amin principle is applicable to itself

You just built another ark, @Daminark?

they are both this guy en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te @AkivaWeinberger

Akiva Weinberger

Then why is one "Vieta's formulas" and the other "Viète's formula"??

11:13 PM

aw i got ninja'd

lol

I KNEW IT

Haha

same people!

Akiva Weinberger

Weird

History is full of errors, DogAteMy.

11:13 PM

well

Vieta is the latin-iz-ation of viete

@Zach: Before you completely forget — we want a polynomial with $g(z)=0\iff f(1/z)=0$. Might that help?

Akiva Weinberger

You know what they say, to air is human @TedShifrin

DogAteMy: Trumplestiltskins is dismantling the EPA, so there won't be air.

Akiva Weinberger

To forgive is the vine.

Whine not?

11:15 PM

@TedShifrin -sqrt ( z^2 +1 )<r < + sqrt (z^2+1)

oh

@Kasmir: First, $r\ge 0$ always.

Akiva Weinberger

Fun fact: Pear, spear, wear, tear, and tear don't all rhyme…

…except in a New Zealand accent, where they all do.

That's cuz English is sooooo logical.

Merry was merry when she was merried.

Akiva Weinberger

They're lucky.

11:16 PM

@Ted Huh?

But otherwise, @Kasmir, yes. Finally.

I pronounce the three words differently

You need to practice setting up triple integrals, @Kasmir.

oh i got it >< 0<r<sqrt ( z^2+1)

Akiva Weinberger

@Daminark In some accents, two or three of "Mary", "merry", and "marry" are pronounced the same.

11:17 PM

Yes I do really :D

I do, slightly, @Daminark. But I grew up in CA and you grew up in TX.

Thanks Ted ! you are the best :D

Well, not exactly

Akiva Weinberger

(Which ones are merged depend on the accent.)

I grew up in NY

11:17 PM

Oh, well, hell.

My parents moved to Texas when I was about 10

And I did high school in Morocco

@AkivaWeinberger where i live Mary and marry are the same, merry is different

So it's weird for sure

Zach, did you read my comment up there???^^^^

Where I live ... oh wait

11:18 PM

no, i didn't sorry

wait

was it the one with

the implies statement?

Akiva Weinberger

I always try to pronounce "ferry" separately from "fairy," even though I wouldn't normally, just because taking a fairy to Staten Island is weird

Yes.

But there might be a cute fairy, DogAteMy :)

But yeah, I say "marry" as "mah-ree", "merry" as "meh-ree", and "Mary" as "m-airy"

Akiva Weinberger

Mah as in the vowel in "mat", right?

You're overdoing the "h"'s, @Daminark.

Akiva Weinberger

11:19 PM

/æ/, /ε/, /ei/, in IPA

I wouldn't put one in the first one.

How do you type that so fast, DogAteMy?

Yeah @Akiva

And yeah wow that was quick

Akiva Weinberger

iPhone. Long press a to get æ, and I have the Greek keyboard on my phone in case of emergencies.

LOL

Ah, I think I can do that on my computer as well

11:20 PM

I hate typing on my phone and iPad.

æ

Option key on the Mac.

It was the Greek that surprised me (since that wasn't a LaTeX symbol).

You can also just hold down a key

Akiva Weinberger

@TedShifrin Long press, also, I think.

Not for $\epsilon$ though

11:21 PM

Oh, cool, @Daminark. I knew not that.

That must be a system 10.? change.

Akiva Weinberger

I think it was.

Paging @Zach. So, now that we're coming up on 2 hours ... ?

@TedShifrin when saying that r start from 0 in that integral seems a bit wierd to me , because from the picture it starts from 1

@Kasmir: No, it does not.

@TedShifrin or never mind i see my mistake now

:)

11:23 PM

Look carefully at your picture.

Good :)

i was looking at the projection when i di dthat analysis thanks again ><

i mean

@Kasmir Do what I do and dodge integral setups entirely, just talk about absolute continuity of measures all day

i notice a pattern within the elementary cyclic sum stuff

drew circle of radius 1 and other of radius sqrt 2 , that what made me think wrong

11:24 PM

@Zach: $g(z)=0\iff f(1/z)=0$. Isn't that an algebraic hint?

like

:P Don't actually do that, it's probably good to actually have practice with being given an integral and doing it

ponders sending Daminark serious homework

What if $g(0) = 0$ ?

We disallow that eventuality.

11:26 PM

@Astyx then the reverse is not the same degree

We specifically said the reciprocals of the roots of $f$ to start with.

idk how im supposed to phrase this

I should go to bed

like whats the relationship between the coefficients?

Have fun with polynomials !

11:27 PM

Bonne nuit, @Astyx.

Akiva Weinberger

No, the bed should go to you @Astyx

Stand your ground

No, @Zach, you already found that. What's a formula for $g(x)$ in terms of $f$?

Akiva Weinberger

(G'night)

Haha good one :p

DogAteMy: You're getting sillier.

Akiva Weinberger

11:28 PM

Is that a bad thing?

Uh, I have to compute tangent spaces of $SL(n,\mathbb{R})$ and $SO(n)$ on this pset. Also we're given that function $f(x,y) = x^3 + xy + y^3 + 1$ and are supposed to find the points $P$ for which $f^{-1}(f(P))$

Is that computational enough?

I'll get back to you on that, DogAteMy.

You didn't finish your sentence, @Daminark. And ... NO.

well $g(z) = f(1/z)$, right? if that means anything

Oh, sorry, for which $f^{-1}(f(P))$ is a smooth 1-dimensional submanifold of $\mathbb{R}^2$.

assuming theyre monic

at least, $f(z)$?

Akiva Weinberger

11:29 PM

@ZachHauk Right, that would work… but can you turn that into a polynomial?

substitute it in?

Akiva Weinberger

With $f(x)=x^3-ax^2+bx+c$, what is $f(1/x)$?

negative-ize all the exponents

Akiva Weinberger

(It's kind of like a polynomial, but with negative exponents)

I've never asked a question quite that way, @Daminark. I sorta like it.

Akiva Weinberger

11:30 PM

@ZachHauk Right. And since we like writing polynomials with the largest exponents on the left, it is…?

Part 2 is to draw the complementary set of these points

$-c + bx^{-1} -ax^{-2} + x^{-3}$

and since we want it to be a polynomial

Akiva Weinberger

@ZachHauk Compare to your answer from way back before

(Before Ted said you should have seen it in a millisecond)

I'm still waiting for the end of that sentence ...

But yeah, I mean there's some amount of computation each pset. We've had to compute solutions to damped harmonic oscillators, specific differential forms, etc

11:32 PM

That was at least 5 milliseconds ago.

ahhhh

Good, @Daminark. You're just missing too much multivariable calculus for my taste. There's so much to do with integrals (and manifolds) and Stokes's Theorem ...

Quadratic forms as well

I did quadratic forms in my course to some extent.

multiply by $x^3$ wouldnt do anything when its 0

and we dont care about any other values than 0

11:33 PM

So $g(x)=x^3 f(1/x)$. Yeah!

We did a bit in order to do spectral theorem

Akiva Weinberger

@TedShifrin Several kiloseconds, but who's counting

That's how you make the "reverse polynomial"!

and so

Yeah, @Daminark, I phrased it that way, too, but you don't have to.

11:34 PM

we pay attention to the negatives, right?

eh stupid question

When I first saw them with Laci, he did bilinear/quadratic forms for their own sake

there are no stupid questions except the one i just asked

It's important to understand $P^{-1}AP$ versus $P^\top AP$, for sure, @Daminark. This is the beginning of tensors :P

Laci had some computation as practice, occasionally because the results were just nice

He's very much a puzzles person in general, I like that style a whole lot

I'm not much of a puzzler.

11:36 PM

Anyways, i must divulge on some consumable foodstuffs, utilizing the cutlery and utensils of various metals which my family unit possesses.

OK, now that Zach's millisecond is up, I'm outta here! Bye, all.

I remember this one problem which was like, you had some pirates on an island

Happy kerfufflery, @Zach.

I wish the same for you

But not "divulge."

11:36 PM

Their map is a polynomial whose roots lead them to the coordinates of the treasure

But it's been ripped

So they only see $x^{17} + 5x^{16} + 13x^{15}$

Prove that not all the roots are real

That was really nice

If I graph the domain $ \{ x,y \in R | x+y > 1 \}$ Shoudnt the domain all the area above a diagonal that goes from x=1 to y=1 ?

Everything above this line desmos.com/calculator/aroqnwa7yp

11:52 PM

Hi all

0

Q: Find $ \max \Im( W(- exp(x)) exp(-x) ) $ for real $x$.

Define the reals $x,y$ as $$ y = \max \Im ( W ( - exp(x)) exp(-x) ) $$ Where $W$ is the standard Lambert W function and $Im$ means the imaginary part. How to find $x$ and $y$ ? Closed forms ( allowing integrals , sums etc ) , contour integrals , numerical methods ?? I know how to express the...

calculus optimization harmonic-functions lambert-w

Any ideas ?

Maybe this is trivial

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