Mathematics - 2018-02-05 (page 2 of 6)

Xander Henderson

03:00

though, to be fair, he is kind of more robot than man

Akiva Weinberger

hums Mario theme

Xander Henderson

YARS!

Mario

Akiva Weinberger

possibly stumbles over the quarter note triplets

0celo7

@Semiclassical who is going to win this

Xander Henderson

Though I went from an Atari 2600 to a Genesis

we never had a NES

0celo7

03:00

@JoeShmo that link does not seem to work

Akiva Weinberger

I haven't ever played Zelda tbh

I'd need a console

Xander Henderson

but I spent hours over at the neighbor's house playing Mario and mother f'n' Metroid

0celo7

I am beyond disappointed @AkivaWeinberger

Xander Henderson

man... Metroid

0celo7

buy a switch now

Akiva Weinberger

03:01

(for the older ones at least, I dunno what the latest one uses)

Xander Henderson

that is damn near the most perfectest game ever

Joe Shmo

@0celo7 try this math.cornell.edu/~sjamaar/manifolds/manifold.pdf

Semiclassical

The thing to note here is that that we’re talking about the hardcore video game scene

Joe Shmo

hm, i dont know why he's hiding some of it. but it works for me

Semiclassical

PS4/XBox etc

Akiva Weinberger

03:02

"Why play when you can watch a Let's Play for free on YouTube?"

Xander Henderson

My genesis is sitting literally three fee from where I am standing right now

but it seems not to work :(

0celo7

@JoeShmo what is the issue

orbit-stabilizer

play*

Antonios-Alexandros Robotis

tetris is pretty fun :P

Akiva Weinberger

It's so strange that there's a whole genre dedicated to the idea that people want to watch other people have fun

0celo7

03:02

@Semiclassical what?

Akiva Weinberger

(I say during the Super Bowl)

Xander Henderson

I don't know if the wall wart is crap, or if the device itself is truely dead

but if my genesis is really dead, I am going to cry

on the other hand, my Kaypro still boots

0celo7

@Semiclassical welp, there it goes

Joe Shmo

@0celo7 problem 2.8 in Reyer Sjamaar's notes on manifolds and differential forms

0celo7

@JoeShmo yes I know

what is your issue with it

Joe Shmo

03:04

hold on, exploring a direction

i'm getting dg = f_1(x1,...xn)dx1 + f_2(0, x2, ..., xn)dx2 + ... + f_n(0, 0, ..., 0, xn) dxn.Mostly would like to know that that's true, before I proceed down that rabbit hole.

Akiva Weinberger

Is the Force from Star Wars fictitious

Like the Coriolis and centrifugal forces

Joe Shmo

next step is to take advantage of the fact that 0 = d alpha = sum_(1 <= i < j <= n) of (df_j/dxi - df_i/dx_j) dx_i dx_j

Akiva Weinberger

(There's a better way to tell that joke)

Xander Henderson

@0celo7 to be fair, I am yanking your chain a little---I know of the existence of the Witcher and Skyrim, but I have never played either

Joe Shmo

i.e. df_j/dxi = df_i/dx_j for all i < j; therein establishing the desired dg = alpha

0celo7

03:06

still shocking

get skyrim

Xander Henderson

I'm in my mid-to-late 30s, married, with a 6 year old daughter, and a thesis to write

who has time for games?

Akiva Weinberger

Depends, is your thesis about games?

orbit-stabilizer

I didn't like skyrim that much

Joe Shmo

@XanderHenderson how's that working out for you? planning on occupying your shoes sooner or later

0celo7

@orbit-stabilizer are you being a contrarian?

Joe Shmo

03:07

(or quite a bit exactly in my mid 30s.....)

0celo7

@XanderHenderson play games with the kid

orbit-stabilizer

Lol no. Is it so surprising that there exist people who don't find Skyrim that fun?

0celo7

I don't have a nice answer foe that

Xander Henderson

the kid likes Journey, which is a rather meditative little game

and she gets a kick out of the original Sonic the Hedgehog games

0celo7

@XanderHenderson I loved pokemon when I was that age

and Mario 3

Xander Henderson

03:08

Yeah, I am too old for Pokemans

they are after my time

0celo7

@JoeShmo ok so did you try the hint?

do it for $n=2$?

Xander Henderson

and, frankly, I don't think that she would have the patience for a JRPG, even one as undemanding as the Pokemans

Joe Shmo

yup working on it @0celo7

but is my dg right @0celo7?

Xander Henderson

she LOVES Race the Sun---her games last about 30 seconds, then she hands over the controller for my turn

Joe Shmo

ill admit, i dont know any calculus :D

Akiva Weinberger

03:10

@XanderHenderson Is there a reason you're spelling it like that

Semiclassical

I did the first two gens of Pokémon

Akiva Weinberger

rather than the correct spelling, ポケモン

Xander Henderson

psh

Pokemans.

0celo7

@XanderHenderson yes, I think so. I see what the point here is. Did you compute $d\alpha$?

Trey

Can anyone help me? I'm trying to calculate $\lim_{x\to \infty} (1+\frac{1}{3}x)^{2x}$. I tried to rewrite it as $$e^{2x\ln(1+\frac{1}{3}x)}$$ but i'm kind of stuck at this point

Xander Henderson

03:10

If for no other reason that it annoys the f*ck out of the youngin's

Akiva Weinberger

(Fun fact, Japanese apparently doesn't have plurals.)

0celo7

@XanderHenderson it does?

Wait, maybe I'm no longer a youngin

it's done

Akiva Weinberger

@Trey Is that $\frac13x$ or $\frac1{3x}$?

0celo7

eagles

Xander Henderson

Did I compute $\mathrm{d}\alpha$? нет

Trey

03:12

@akiva $\frac{1}{3}x$

0celo7

I meant @JoeShmo sorry

@JoeShmo did you compute $d\alpha$?

Joe Shmo

huh?

0celo7

$d\alpha=0$

Akiva Weinberger

@Trey Then that's just $\infty^\infty=\infty$

0celo7

this gives a condition on the $f_i$'s

Joe Shmo

03:13

yuh yuh..

Xander Henderson

$\infty^\infty = 2$...

Joe Shmo

so 0 = d alpha = sum_(1 <= i < j <= n) of (df_j/dxi - df_i/dx_j) dx_i dx_j

Xander Henderson

come on... everyone knows that !

GAH! MATHJAX THAT SHIZZLE!

orbit-stabilizer

$\int_{a}^{b}fd\alpha = inf \sum M_i \Delta \alpha_i$

0celo7

@orbit-stabilizer not helpful

Akiva Weinberger

03:14

I need to play Undertale at some point

Trey

@AkivaWeinberger that can't be... according to my textbook it is $e^{\frac{2}{3}}$

0celo7

@Trey then you made a typo

Akiva Weinberger

@Trey And you're sure you didn't make a typo? (Or perhaps the textbook did!)

$\lim(1+\frac1{3x})^{2x}=e^{2/3}$, I believe

Trey

@AkivaWeinberger you're right, it's actually $\lim_{x\to \infty} (1+\frac{1}{3x})^{2x}$

Akiva Weinberger

All right

The approach you started will work, but it's annoying

A simpler way is to try to relate it to $\lim_{n\to\infty}(1+\frac1n)^n=e$

orbit-stabilizer

03:17

How will he deal with the 3?

0celo7

what's the proof of that, anyway

Akiva Weinberger

Specifically, write $x=n/3$ in the textbook's problem

0celo7

@AkivaWeinberger

Akiva Weinberger

and substitute

0celo7

I don't think I've ever seen it

orbit-stabilizer

03:17

ah

0celo7

I know the corresponding result in spectral theory, but, well

I think it uses the real valued result

Akiva Weinberger

@0celo7 It depends on your definition of $e$

Some people use it as the definition

Trey

@AkivaWeinberger I thought about that, but how can I relate $3x$ with $2x$?

0celo7

@AkivaWeinberger the correct definition is clearly the series

Akiva Weinberger

All you need to do is prove that the various definitions refer to the same number

@Trey What happens when you do the substitution?

0celo7

03:18

@AkivaWeinberger yes, obviously

Joe Shmo

@0celo7 i dont know how to compute the partial derivatives of f_i

0celo7

List of representations of e

The mathematical constant e can be represented in a variety of ways as a real number. Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other sort of limit of a sequence. == As a continued fraction == Euler proved that the number e is represented as the infinite simple continued fraction (sequence A003417 in the OEIS): e = [ 2 ; ...

Akiva Weinberger

@0celo7 There's some trickery with uniform convergence and stuff, but just expand $\lim(1+\frac1n)^n$ binomially

0celo7

@AkivaWeinberger that makes sense

@JoeShmo you don't need to. I haven't worked it out but the idea is that $d\alpha=0$ gives a condition on the $f_i$, like maybe $f_i$ is constant in some of the variables

and then you tie it into your formula for $dg$

Daminark

@AkivaWeinberger yeah def do so, it's excellent

Akiva Weinberger

03:22

a wild daimrank appears

${}/S_n$

@0celo7 Have I told you my favorite definition of $e$?

It's the only real number satisfying $e^x\ge x+1$ for all $x$

(Positive and negative $x$)

I like it because it does not involve any calculus

0celo7

is it obvious why it's the only number satisfying that?

Akiva Weinberger

and you can prove $\ln2=1-\frac12+\frac13-\dotsb$ just using that definition and no calculus

(well, the last step is the sandwich theorem, but other than that)

0celo7

@AkivaWeinberger how is 1-1/2+...supposed to make sense without limits

Akiva Weinberger

The last step is the sandwich theorem

I forget the exact details, but I think you show $\ln2-\frac1{2n}<1-\dotsb\pm\frac1n<\ln2+\frac1{2n}$ or something like that, without calculus

0celo7

I wrote my message before yours appeared

my internet is being a derp

@AkivaWeinberger but is it obvious that such a unique number exists?

Trey

03:28

@AkivaWeinberger how will I do a substitution if the denominator and the exponent are different?

Xander Henderson

The only "sandwich theorem" that I acknowledge is the "ham sandwich theorem." The other, lesser theorem is the "squeeze theorem."

0celo7

@XanderHenderson That's likely what he was referring to.

Akiva Weinberger

@Trey Take $\lim(1+\frac1{3x})^{2x}$. Write $x=\frac n3$. What do you get?

@XanderHenderson What about, Theorem: Sandwich is tasty. Proof: Om nom nom yes QED

Antonios-Alexandros Robotis

lol @XanderHenderson

Akiva Weinberger

@0celo7 No

Xander Henderson

03:32

Theorem Theorem: If if, then then.

Akiva Weinberger

@0celo7 Well actually I think proving there's at most one number satisfying it isn't that hard

0celo7

@AkivaWeinberger my favorite definition is $f(1)$, where $f$ is the unique solution of $f'=f, f(0)=1$

Akiva Weinberger

though nonobvious as well

0celo7

@AkivaWeinberger proof?

Xander Henderson

Proof: Proof.

0celo7

03:33

Proof: Clear

Joe Shmo

@0celo7 all i can get from the n=2 example is df_2/dx1 = df_1/dx2;

Akiva Weinberger

You probably sub $x$ for $-x$ and move stuff around to get an upper bound, and then sub $x$ for $x\ln a$ to get similar bounds for $a^x$, and then shove stuff together and hope it works

I haven't worked it out

Also, if you sub something into $e^x\ge x+1$, you can get that it's equivalent to showing that $x^x$ has a minimum at $x=\frac1e$ (or that $\sqrt[x]x$ has a maximum at $x=e$)

(I think you sub $x\mapsto\frac xe-1$ into it? Something like that)

0celo7

@JoeShmo ok the game is over so I'll go back to my ~~cave~~ desk and work it out

Joe Shmo

I'll take it you were an eagles fan if you're in such a good mood?

0celo7

I was neutral

Joe Shmo

03:38

me too. except i was rooting for the eagles. because all my friends are from boston.

0celo7

your dg has to be wrong

Joe Shmo

how?

0celo7

I'll write the solution, one sec

Xander Henderson

man, eagles are awesome

there are some great videos of them ripping drones right out of the sky

0celo7

the crossed out stuff was in error

you can probs figure it out from here

Joe Shmo

03:49

are you a physicist?

0celo7

no

Joe Shmo

how are you computing dg?

0celo7

$dg=\partial_ig \, dx^i$

Joe Shmo

the sum over all such partials?

0celo7

yes, always use Einstein notation

Semiclassical

03:57

and you say you're not a physicist :P

0celo7

@Semiclassical every serious geometer or analyst understands Einstein notation

Joe Shmo

@0celo7 are you a geometer

0celo7

yes

Joe Shmo

did model theory ever come in handy, and what's your favorite tools from topology?

0celo7

I have no idea what model theory is

Semiclassical

04:00

mathematical logic stuff

0celo7

Arzela Ascoli is the best result from topology

Joe Shmo

hrushovski, shelah, zielber.

0celo7

never heard of them

Joe Shmo

ok. so d1f2 is df2/dx1?

0celo7

yes

Joe Shmo

04:02

@Semiclassical im trying to understand whether model theory is nonsense, or there is sense

Trey

@AkivaWeinberger that worked, thank you!

Semiclassical

i have no real reason to suppose it's nonsense. but it being sensible doesn't mean it's relevant/useful for most people

Joe Shmo

@0celo7 how come the partial commutes with the integral?

0celo7

@JoeShmo do you know the theorem for integration under the integral sign?

Joe Shmo

something something leibniz something else

0celo7

04:04

oh I thought you were asking something more delicate

when everything is smooth, you have

Leibniz integral rule

In calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form ∫ a ( x ) b ( x ) f ( x , t ) d t , {\displaystyle \int _{a(x)}^{b(x)}f(x,t)\,dt,} where − ∞ < a ( x ...

Joe Shmo

yuh, but x1 is not a constant?

0celo7

that formula has variable bounds

it works for what I did

(but you don't really need that strong formula)

@Semiclassical turns out I don't need the Pohozaev identity after all .-.

Joe Shmo

@0celo7 why is integral 0 to x1 of d1f2(0,t) dt = 0?

0celo7

@JoeShmo the derivative of that integral wrt. x_1 is zero

f_2(0,t) is constant in x_1

Joe Shmo

yes, why?

how do we know?

0celo7

04:14

it's evaluated at zero

Joe Shmo

ah!

dur

Semiclassical

@0celo7 lool

Joe Shmo

very nice. ill try to take it from here. ill update tomorrow.

Semiclassical

well, still good to have found that review on it

0celo7

@Semiclassical I decided to not prove what I wanted to prove. I can make due with a little weaker result and it's not worth 20 pages to save 2 lines in another proof in the main text

(seriously, the proof I was going to attempt takes 20 pages)

(I did not know that at the time)

Semiclassical

04:22

it'd really be convenient if we knew in advance how long stuff like that would take

0celo7

@Semiclassical I still want to understand the proof, but it doesn't need to go in the thesis

maybe a 1-page summary of the proof

Semiclassical

right

Daminark

05:35

Here's a good problem

Antonios-Alexandros Robotis

?

Daminark

Compute the coefficient of $x^7$ in the polynomial $(x-a)(x-b)\ldots (x-z)$

You're free to use any tools you want, Vieta, etc

Cookie Toast

05:59

There's gotta be some relatively simple way to do so using binomial coefficients @Daminark

Daminark

Go for it and see

Ted Shifrin

Demonark: Are you being your usual self with this question?

Hi @Cookie

Daminark

What do you mean "usual self"?

Ted Shifrin

I mean the answer is 0.

Daminark

Lol you spoiled it, I dunno it's kinda fun

Ted Shifrin

06:14

Nah, it's really not.

Daminark

Someone gave this to me and I was trying to do stuff for a few minutes and was like oh wait I've been bamboozled

Ted Shifrin

It's the reason I let $a,b,c$ be constants, but $u,v,w,x,y,z$ are always variables. :)

Daminark

shrug fun is in the eye of the participant, I suppose

Ted Shifrin

Well, I thought about it because I was trying to decide if it would be the $(26-7)$th symmetric function of all the letters ...

Daminark

Nice

Ted Shifrin

06:15

Right. So the answer is that yes, you were being your usual self. "fun"

Mambo

Hi

Daminark

This is true

Ted Shifrin

@Cookie has more worthwhile things to waste his time on ;)

hi Mambo

Daminark

I don't have quite that many actual good problems to float around so rip

Find all solutions to $y^2 = x^3 - 2$? That's a fun one

(solutions in $\mathbb{Z}$ ofc)

Mambo

Trying to think of a continuous function from $$l_\infty$$ to $$c_0$$ that restricts to identity on the lattet

*latter

Ted Shifrin

06:18

Don't use double dollar signs, please. Single will be fine.

What's $c_0$, again?

Mambo

Set of all those sequences that converges to zero

Daminark

Wait that's a closed subspace, right?

Yeah it totally is

Mambo

There is no such linear function

Cookie Toast

@ted Hey! Cookie has an essay on government you should be working on. And he's also reading this textbook by some Ted guy or something ;)

Mambo

If there is any linear function, it would mean $c_0$ is complemented in $l_\infty$ which is not true

Ted Shifrin

06:22

Does your essay talk about how the president is destroying the structure established by the Constitution, with the Congress as accomplices?

Daminark

Ah rip

Cookie Toast

Actually, I'm making an argument about how our country is subdividing ourselves into micro-communities of those who won't present strong challenges to our viewpoints. This leads to inaction, and we're all just as complicit in the presidents actions. I.e. my inaction against his action is just as bad as his action. We're reading Thoreau right now, so its a good tie-in

Ted Shifrin

Yes, that's a definite development over the past decade or two.

Good going.

Cookie Toast

Yeah, the first essay is probably going to be the easiest if only because there's so much to write about on the topic of government. The rest of the semester's going to be a bit more difficult

Either way, its kind of ironic, cause I'm writing an essay arguing that inaction is a form of support to the president, and here I am being relatively politically inactive day to day. I've found it's a good chance to self reflect

Ted Shifrin

I'm truly glad you're taking it so seriously. I wish more of your fellow students and citizens would.

Mambo

06:32

What degree do you study @CookieToast?

Cookie Toast

@Ted: To be fair to our fellow citizens, and this includes myself; it's much more uncomfortable to write an essay about the current political climate. I have to choose stuff like that though, because otherwise I'll never get the inspiration.
I study math @Mambo

Ted Shifrin

We're quickly working on changing @Cookie's mind on that major, though, Mambo.

:P

Cookie Toast

Impossible. How could I be anything else than a math major, when I have great ideas like rotary-dial-telephone vector notation?

Ted Shifrin

Say what?

Cookie Toast

@Ted Ah, sometime a while back on here I was suggesting a new vector notation that involved writing each component in a circle, similar to the rotary-dial telephones, instead of as a column. Some people took me seriously too :P

Ted Shifrin

06:40

Thankfully, I was spared that "inspiration."

Cookie Toast

I was actually considering writing about that legislative push back in the day to legally change the definition of $\pi$ to $\frac{22}{7}$ for "the sake of learning" or something, but in the end it didn't feel as strong of a topic compared to what I could write about with today's politics.

Ted Shifrin

07:06

@Mambo: What about subtracting the limsup of the sequence? That shoukd be continuous but definitely not linear.

Oh, no, that doesn't map everything to $c_0$. Blah.

Cookie Toast

@Ted are you an advocate of the "skip around and look at what interests you" approach when self studying, or more of the "linear progression through the text" train of thought?

Abcd

Show that $(x^2+y^2)^4 = (x^4 - 6x^2y^2+y^4)^2+(4x^3y- 4xy^3)^2$

Ted Shifrin

If you're in $c_0$, leave it alone. Otherwise map to the sequence $(\limsup a_n)/n$.

Abcd

I don't want to expand the whole thing and show :\

Ted Shifrin

Depends if you're trying to master it or just pick up things here and there, @Cookie.

Abcd

07:18

Any method to solve this hard problem?

Ted Shifrin

I don't see an alternative, @abcd, but certainly factor out $(4xy)^2$ from the second term.

Abcd

@TedShifrin Did that...wasn't helpful.

Cookie Toast

@Ted with a lot of stuff, i'm not worried about complete mastery, but I do want to just forget everything. I'm reflecting back to my study methods over break last semester, and I feel like I was a little too strict with my approaches

Abcd

Its a question in the chapter "complex numbers"

Note that $x^2+y^2 = z\bar{z}$

I tried to use this too. Note: $z= x+iy$

Cookie Toast

That notation means $z$ and $z$'s conjugate, yes?

(a+bi)(a-bi)?

Abcd

07:21

yes

Cookie Toast

So try factoring out the term Ted suggested. Do you notice anything about the second term ?

Ted Shifrin

So the second term looks like the imaginary part of $z^4$ and the first term is the real part. Done.

Each squared, of course.

Abcd

@TedShifrin wait, no.

Ted Shifrin

Good hint — the chapter name.

The lhs is $|z^4|^2$, of course.

Abcd

$(z\bar{z})^4 = ((z\bar{z})^2 - 8x^2y^2)+ ((4xy)(x^2-y^2))^2$

@TedShifrin ikr

then?

Ted Shifrin

07:27

Forget about conjugates.

Abcd

$|z|^8 = (|z|^4 - 8x^2y^2)+ ((4xy)(x^2-y^2))^2$

Ted Shifrin

Huh?

Read what I said above.

Abcd

@TedShifrin $|z^n|= |z|^n$

Ted Shifrin

I know. I'm not a dummy.

Abcd

@TedShifrin i have eliminated conjugates.

Sorry.

Ted Shifrin

07:29

Real and imaginary parts of $z^4$ ?

Abcd

@TedShifrin Real part is $\dfrac{z^4 + \bar{z}^4}{2}$

imaginary part is $\dfrac{z^4 - z^4}{2i}$

Ted Shifrin

Grrr ... in terms of x and y!

Abcd

is there a shortcut to find that?

or am I supposed to expand ^4?

Ted Shifrin

You need binomial expansion, but I did it in my head in 2 seconds.

Abcd

Imaginary part is x.

Ted Shifrin

07:38

Huh?

Abcd

Now its correct I hope.

Ted Shifrin

What? You are taking $(x+iy)^4$?

Abcd

yes

Ted Shifrin

Every term has degree 4!

Abcd

Imaginary part is $x^4$, surely.

Ted Shifrin

07:42

Oh come on. Write out all 5 terms.

I'm losing patience.

Abcd

I used this property: Imaginary part is $\dfrac{z-\bar{z}}{2i}$

Let me try again then.

Ted Shifrin

Imaginary part will be the terms with $i$.

What's the real and imaginary parts of $2+3i$?

Abcd

Real: 2

imaginary 3

Ted Shifrin

Now do that in the polynomial expansion.

Abcd

Imaginary part is $4x^3y- 4xy^3$

Ted Shifrin

07:47

Not quite.

Write down the 4th row of Pascal's triangle.

Right.

Abcd

Real part is $x^4 -6x^2y^2 +y^4 $

Ted Shifrin

Done.

Alessandro Codenotti

Hi chat

Abcd

@TedShifrin Done :)

Ted Shifrin

Hi demonic Alessandro.

Mambo

07:51

@TedShifrin do you think this is Lipschitz continuous?

Ted Shifrin

What's the norm on $c_0$?

Sup?

Mambo

Subspace metric

Ted Shifrin

Yeah, I bet it is. Ask Alessandro if he believes this ...

Alessandro Codenotti

What's the question?

Daminark

Can you have a continuous function $\ell^{\infty} \to c_0$ which restricts to the identity on $c_0$

Mambo

07:58

Thank you @Daminark. I was struggling to type it in mobile

Daminark

Now, Ted came up with a function by sending everything not in $c_0$ to $(\limsup a_n)/n$

Ted Shifrin

Do you believe my function, Demonark?

Transcript for

$Mathematics$ Mathematics