Mathematics - 2017-10-19 (page 1 of 4)

Akiva Weinberger

00:02

In other words, $\dfrac{n^3+100}{n+10}=n^2-10n+100-\dfrac{900}{n+10}$, and the problem states that the left-hand side is an integer

so the right-hand side is an integer as well, which means $\dfrac{900}{n+10}$ must be an integer, meaning that $n+10$ divides $900$.

Leaky Nun

@AkivaWeinberger thanks :P I was washing my sock and as I came back you have already answered lol

I would still type my explanation here though

Akiva Weinberger

To be clear, this is a consequence of the fact that $(n+10)(n^2-10n+100)=n^3-1000$, which itself is a consequence of the difference-of-cubes equation

$a^3-b^3=(a+b)(a^2-ab+b^2)$

Leaky Nun

We know that $n+10$ divides $n^3+100$ and that it divides $(n+10)(n^2-10n+100)$, so they have to divide their difference, namely $900$

Akiva Weinberger

Wait, I misspoke

Leaky Nun

lol

$a^3-b^3=(a-b)(a^2+ab+b^2)$

Akiva Weinberger

00:07

$(n+10)(n^2-10n+100)=n^3\color{Red}{{}+{}}1000$, which is a consequence of the sum-of-cubes equation

$(a+b)(a^2-ab+b^2)=a^3\color{Red}{{}+{}}b^3$

Leaky Nun

oh, I was also wrong lol

my identity was right but it didn't match

(ok strictly speaking it can match but whatever I'm going to end this argument with myself)

Akiva Weinberger

The answer is still 890

Akiva Weinberger

00:21

So Google made a Go bot that thought itself how to play Go without learning from any human moves

AlphaGo Zero

Tyler L

Does this solve the sleeping beauty equation? ((n-1)/n)*(1/3) + (1/n)*(1/2)

If n = 1, the probability should be 1/2

Akiva Weinberger

and it beat both previous versions of AlphaGo (which beat the human champions Lee Sedol and Ke Jie), 100–0

@TylerL It definitely starts at 1/2 and goes to 1/3 in the limit, but why that equation specifically?

Tyler L

I'm trying to make the second coefficient = 0, but didn't know how. It makes sense that if n-1 is the first coefficient... the second coefficient should be that leftover possibility (1 out of n = 10 chance that you flip HHHHHHHHHH)

wait

n-1/n works but not sure 1/n works

should it be 1/n! ?

1 combination in n combinations = all heads

so the sleeping beauty problem can be modeled ((n-1)/n)(1/3) + (1/n!)(1/2)

Akiva Weinberger

1/2^n maybe?

Chance of n flips all being heads is 1/2^n

Ted Shifrin

hi, DogAteMy

Akiva Weinberger

00:28

Hiya

Tyler L

nice

makes the equation look cool as well. ((n-1)/n)(1/3) + (1/2^n)(1/2)

@AkivaWeinberger that makes sense now that I remember learning binary is 2^bits

Akiva Weinberger

I don't think it makes sense for the probability to change as n changes… the only problem is how exactly we measure the probability. If we do this on n separate weeks and she always answers Heads, is the probability that Heads is correct (a) the number of days on which she's right over the total number of days, or (b) the number of weeks on which she's right over the total number of weeks?

Tyler L

Should it be 2^(n-1) though? So if n === 1, you get 1/1 * 1/2

Akiva Weinberger

(a) would give 1/3, (b) would give 1/2

Then again, you've thought about this problem longer than I have

Tyler L

Idk Akiva

lol still thinking about what you said

You're measuring how certain she is that it's 1/2 I guess... 1/2^(n-1) works because she should be exponentially less certain after a few experiments

Akiva here's the simulation after 1000 tries (it nears 1/3) codepen.io/TylerL-uxai/pen/Bwvrwv

Leaky Nun

00:37

@AkivaWeinberger do they play with themself?

@AkivaWeinberger beating is not a transitive relation :P

Tyler L

shoot

If you put 1 in for n, it's 2^0 = 1 so it's 1/1 * 50%

Akiva Weinberger

@LeakyNun Yeah, it's based on self-play.

(Er, "does it play with itself")

Leaky Nun

@AkivaWeinberger nice

Akiva Weinberger

I assume at some point they'll let it play against Ke Jie or whatever

Ted Shifrin

DogAteMy: I think this needs to be X-rated.

Akiva Weinberger

00:46

Dirtiness is in the mind of the beholder. :P

Ted Shifrin

Well, and what's your point?

Akiva Weinberger

Maybe I should learn how to play Go at some point. But it's considered harder than chess, and I'm still pretty crappy at chess

Ted Shifrin

Go, I'm told, is more topological, but, yeah, I'm horrid at chess, so I avoid both.

I stick to bridge.

Akiva Weinberger

Yeah, you try to enclose areas and such

or something

Ted Shifrin

Something to do with $\pi_0$, too.

Tyler L

00:54

(((2^n)-1)/2^n)*1/3 + 1/(2^(n-1))*(1/2)

Eric Silva

@Akiva i play go and I'm much worse at chess than go despite liking both a lot

Akiva Weinberger

Hm

Have you looked at the AlphaGo games?

Ted Shifrin

heya Eric

Akiva Weinberger

(Hlello)

Eric Silva

I watched all the Lee sedol games

Ted Shifrin

00:56

DogAteMy, apps done?

Meow Mix

howdy

Eric Silva

Also hi everybody lol

Ted Shifrin

hi Meow

Meow Mix

im finally getting around to readng spivak

does anyone have an interesting limit problem

also ted im gonna email you something one sec

Akiva Weinberger

@TedShifrin Mostly, gonna do one final check with the college advisor

Well, for the common app anyway, and for the Yale-specific questions

Meow Mix

00:57

where did you apply? if you dont mind my asking

Ted Shifrin

Cool, DogAteMy

Akiva Weinberger

Yale for Early Action, gonna do a bunch of less-selective and safe schools if that doesn't work out

Ted Shifrin

@Meow, have you done limits of rational functions yet?

Meow Mix

i dont think so, can you give an example problem?

Ted Shifrin

I wouldn't say they're less selective and safe, DogAteMy, unless you revamped your list completely.

Easy one would be: $\lim_{x\to 3}\frac{x-2}{x+3}$. Or try using quadratics, etc.

Akiva Weinberger

00:59

The ones you gave me?

Ted Shifrin

DogAteMy: Well, some of the ones you had in the first place.

Akiva Weinberger

$\lim\frac{4x^4+1}{2x^2+1}$

Meow Mix

one thing that's kind of iffy is like

i know how to prove a limit using epsilon-delta

Akiva Weinberger

@TedShifrin Yeah, those were pretty selective

Ted Shifrin

Oh, that's too yucky.

Meow Mix

01:00

but i dont have any tools yet

so i cant really tell what the limit is gonna be

Ted Shifrin

There are methods, @Meow. Which edition are you reading?

Meow Mix

also, did you mean $x-3$?

Ted Shifrin

Well, you can sure guess the limit.

Meow Mix

in the denom

Ted Shifrin

No.

Akiva Weinberger

01:01

A very powerful tool is that $\lim f(g(x))=f(\lim g(x))$ if $f$ is continuous.

Meow Mix

everywhere continuous?

Ted Shifrin

Meow doesn't know officially what continuity is.

Akiva Weinberger

For example, $\lim\dfrac1{g(x)}=\dfrac1{\lim g(x)}$

Ted Shifrin

These things follow from limit theorems.

Akiva Weinberger

and $\lim e^{g(x)}=e^{\lim g(x)}$, etc

Meow Mix

01:02

dont have mathjax one sec

Akiva Weinberger

Lim 1/g(x)=1/(lim g(x))

Ted Shifrin

@Meow: Plus you haven't told me which edition.

Meow Mix

sorry let me check

Akiva Weinberger

lim e^g(x)=e^(lim g(x))

Ted Shifrin

These are theorems, DogAteMy. He needs to prove them.

And Spivak doesn't get to exponential until integration.

Tyler L

01:03

What about something like this for the sleeping beauty problem? i.imgur.com/xQqffH5.png

Akiva Weinberger

@TedShifrin Right, sure. Once he proves it, it's a very powerful tool.

Meow Mix

apparently, third

Akiva Weinberger

@TedShifrin Hrm

Ted Shifrin

Ah, OK. I added some explicit stuff on limits for the fourth, @Meow. Let me email you a handout I gave my classes.

Meow Mix

and let me email you

Akiva Weinberger

01:04

@TylerL Right, that's equivalent to the last thing you wrote

Tyler L

I dropped the * 1/2 and just did 1/2^n

It works for case n = 1 but idk how to prove it with induction or anything

Akiva Weinberger

Yeah but the last one had 1/2^(n-1)

Tyler L

yeah

Akiva Weinberger

so the 2 and the 2^(n-1) combine to make 2^n

Meow Mix

email sent

Tyler L

01:05

hahaha

smart

Ted Shifrin

mine too, Meow

Akiva Weinberger

@MeowMix If you have a graphing calculator (desmos), try graphing $\frac{3x^2+1}{2x^2+1}$, try changing the exponents and coefficients, see if you can make conjectures

Eric Silva

@Ted Spivak does exponential that late??? wow i forgot he does some weirdness

Ted Shifrin

Oh, he knows this kind of stuff, DogAteMy. He's trying to learn Spivak now.

Yeah, Eric, log defined as a definite integral. It's really the only rigorous thing to do.

Tyler L

it works for n = 999999999 and for n = 1, but idk about proving it for n > 1

Akiva Weinberger

01:06

Oh, I see. So he just needs to do it rigorously.

Eric Silva

i mean power series tho

Tyler L

maybe run it through the sigmoid function?

Ted Shifrin

Actually, Eric, before the "early exponentials" texts appeared, most calculus books did that — Thomas, etc.

Power series comes way after integrals.

We're not doing Rudin.

Eric Silva

lol

Akiva Weinberger

Rudin did $a^x$ pretty ea—sniped

Meow Mix

01:07

did you send it to the right e-mail

Eric Silva

it's been a long time since i picked up a calculus text

Ted Shifrin

Oh, did you switch emails.

Meow Mix

yes, it's the last one you used when you sent the CG

Ted Shifrin

gmail addy

Akiva Weinberger

—rly, and then defined $E(x)$ as a power series, and proved it equals $e^x$ where $e=E(1)$

Meow Mix

01:08

or alternatively, the one i just sent a email to you withj

Akiva Weinberger

[email protected]

(joking)

Ted Shifrin

Oh, I have never seen that addy before.

Meow Mix

i thought you said zach attack but it turned out to be attach

zatch attatch

^ pronunciation

cba to write out the IPA

Akiva Weinberger

Cba?

Meow Mix

cant be asked

@TedShifrin we have an issue

Ted Shifrin

01:10

I resent, Meow.

LOL?

Meow Mix

your email doesnt have the pdf as an attachment, it's just text that says <limits.pdf>

Akiva Weinberger

/kbə/

Eric Silva

I guess in my brain sequences and series belong together and these are more foundational than the derivatives and integrals but this sounds like bad pedagogy in a calc book

Ted Shifrin

The attachment didn't go?

GRR.

Meow Mix

These kids and their newfangled electronic mail!

Akiva Weinberger

01:11

/gr̩ːː/

Ted Shifrin

Sent again. GRR.

Akiva Weinberger

/ˌgəɹəɹə/

Ted Shifrin

DogAteMy: Next year you'll miss me :D

Akiva Weinberger

Next year I can still go on chat, no?

Is something happening on your end?

Ted Shifrin

Nah, you'll be way too busy.

Meow Mix

01:13

he's moving to the indigenous forests

Eric Silva

college sucks

Meow Mix

oh

Akiva Weinberger

Ahh

Ted Shifrin

Meow, did you get it?

Akiva Weinberger

I'm actually gonna take a gap year in Haifa first

Meow Mix

01:14

affirmative

Ted Shifrin

Oh really, DogAteMy? Interesting.

Eric Silva

i feel like i know a lot of people who took gap years in israel

Akiva Weinberger

but I'm probably gonna take engineering there (Technion - Israel Institute of Technology)

Ted Shifrin

Eric, all Jews.

Akiva Weinberger

@EricSilva Extremely common in my community. All three of my siblings did it

Ted Shifrin

01:15

DogAteMy: Interesting. You strike me as very un-engineering, but I think that's a great idea.

Akiva Weinberger

though they all went to Jerusalem

Meow Mix

oh P.S. that code hs no comments, sorry

Akiva Weinberger

@TedShifrin Yeah, I know nothing about it, but I might find it interesting

In any case, useful to learn anyway

Eric Silva

I lived in Boca Raton in soflo which had loads of new york jew expats so that's probably why i know a lot of people who did it

Ted Shifrin

A change of pace from your theoretical math profile, sure, DogAteMy.

Eric Silva

01:16

everyone i know loved it so sounds like ull have a good time @Akiva

theoretical math = worst math

Ted Shifrin

?

Akiva Weinberger

Get'im!

Also I think they prefer to call it "pure math" :P

Ted Shifrin

they?

Eric Silva

im being mostly facetious

Ted Shifrin

I do not like the "pure" moniker.

Akiva Weinberger

01:17

for it is without blemish

Ted Shifrin

It suggests that applied math is impure, and I really don't like that.

Eric Silva

i like theoretical vs applied

Akiva Weinberger

Tainted by the physical world

Eric Silva

but i dont like the rigidity of the nomenclature

Ted Shifrin

And it really isn't vs ... there is lots of overlap.

Eric Silva

01:19

i like the idea of being able to straddle across the "divide"

Ted Shifrin

One of my favorite courses I taught was an applied math year-long class.

But we did some proofs and all sorts of neat stuff.

Akiva Weinberger

And at the end of the course, you built a dam

Meow Mix

i have a question ted

Ted Shifrin

Damn you, DogAteMy, we didn't.

Meow Mix

how did they make these kinds of graphs back then?

Eric Silva

01:20

ill probably apply to at least one applied reu for my last undergrad summer

Meow Mix

like $\sin \frac{1}{x}$

Ted Shifrin

I actually don't know that, Meow. Graphic artists had serious jobs.

Mainframes could do graphics.

Meow Mix

i wish i owned a 1970s mainframe

Ted Shifrin

It would be as big as your parents' house.

Meow Mix

ok

maybe in my backyard

Akiva Weinberger

01:21

@TedShifrin Well that's a dam failure

Meow Mix

wait could i even fit another house in my backyrd

Ted Shifrin

thanks, DogAteMy

Meow Mix

what did i just watch

Akiva Weinberger

A… a dam failure

Secret

[Random]
Leaky's mention of polynomial division in the previous conversations on solving $n+10$ divides $n^3-100$ once again trigger this attempt again:

$\sin a = \frac{b}{c}, \cos a = \frac{d}{f}, \tan a = \frac{bf}{cd}$

$ce^{2ia}-c=2b,fe^{2ia}+f=2d,(bf-icd)e^{2ia}+(bf+icd)=0$

Suppose $e^{2ia}=q\in \Bbb{Q}$. Then

$c(q-1)=2b,f(q+1)=2d,(bf-icd)q+(bf+icd)=0\implies bf(q+1)+cd(1-q)i=0 \implies bf(q+1)=0 \wedge cd(1-q)=0$

Equation 1 and 2 may have countably many solutions as the constraints are $q=\frac{2b}{c}+1, q=\frac{2d}{f}-1$

Eric Silva

01:22

@Akiva idk why but watching this really disturbs me

Akiva Weinberger

Or a model of one

Meow Mix

oh speaking of mainframes

im probably gonna buy myself a used server

Akiva Weinberger

Soil got liquified

Ted Shifrin

Anyhow, I'm about to go cook dinner. Meow, did you have a question about what I sent?

Eric Silva

@Ted what's for dinner

Meow Mix

01:24

umm no i dont think so

Le petite rolleux de Pizza

Ted Shifrin

I cooked some coho salmon last night, so a bit of leftover plus red dandelion greens plus ... something else.

OK, well, y'all take care. Talk later.

Eric Silva

have fun

Akiva Weinberger

A country was punished to have a record number of dams forever

Eternal dam nation

Eric Silva

h o r r i b l e

Akiva Weinberger

Aw, my sink is clogged

Daminark

01:26

Rip in sink

Akiva Weinberger

Hi dam-in-ark

Eric Silva

i blame Daminark for that one

Meow Mix

ok so

Eric Silva

he shouldnt have made his name so punnable

Daminark

My real name is even worse so...

Eric Silva

01:29

what do you a-mean

Akiva Weinberger

You sound-a Italiano right-a there-a

Silv-a

Eric Silva

lol

Meow Mix

can i separate the limit of a ratonal funtion into two limits, the numerator and denominator?

wait no what if the denominator is 0

tonytouch

Hello. Can anybody help me prove the correctness of this algorithm: imgur.com/a/aHPiZ

Eric Silva

@Daminark shame you left rep theory as soon as it turned into full on number theory + group theory

Akiva Weinberger

01:35

If $\lim g(x)\ne0$ and $f,g$ are continuous, then $\lim\dfrac{f(x)}{g(x)}=\dfrac{\lim f(x)}{\lim g(x)}$

Meow Mix

ok

so i cant use that here

then what the hell am i supposed to use

Akiva Weinberger

@tonytouch Tryna show a is a multiple of b?

Meow Mix

all i have to work with is this theorem

no calc tools

0celo7

@MeowMix what exactly do you want to do?

Akiva Weinberger

Don't know much about that particular programming language but the idea seems to make sense

Meow Mix

01:37

0celo7

which one?

Meow Mix

number 6 without calc tools

only limits

and i have to use theorem 2

0celo7

do you know what a conjugate is?

Akiva Weinberger

Factor them

$x^2-1={}?$

Meow Mix

how does one factor a square root

0celo7

01:39

that's definitely not the right term for what we're suggesting...

Meow Mix

sorry akiva, meant problem 6

0celo7

more like reverse factoring

Akiva Weinberger

Heh

Meow Mix

no because akiva said factor

Akiva Weinberger

Yeah I know what you do there

Meow Mix

01:39

then i realized he was looking at the wrong one

Akiva Weinberger

You don't factor but you do something similar

Eric Silva

rationalize the numerator?

tonytouch

@AkivaWeinberger before this I had to prove by induction that x_n + by_n = a, where x_n an y_n are the values of the variable of x and y after n iterations. I think I had to use this to prove the correctness of the algorithm

Akiva Weinberger

@0celo7 Is your icon a double torus with a disk removed?

Topologically

0celo7

No, random

What's special about a double torus with a disk removed?

Akiva Weinberger

01:40

Well it's clearly a genus-something torus with a disk removed

All orientable surfaces with one boundary are of that form

Seems to be if you seal off that boundary with a (topological) disk you end up with a genus-2 torus

I only mention it because I spent a lot of time thinking about a version of that shape (except made of flat pieces instead of one curve)

Meow Mix

wait a second

wait no that doesnt lead anywhere

then i just have another square root expression

0celo7

Are you sure it will only have two holes?

Akiva Weinberger

That's the tricky visualization part of the puzzle. But I think so

Theoretically we could just calculate the Euler characteristic but that seems annoying

Meow Mix

wait a second

this does lead somewhere

can someone confirm my suspicion

0celo7

@MeowMix Did you not learn how to rationalize denominators?

Meow Mix

01:44

i learned denominators

is it, multiply by $\sqrt{a+h} + \sqrt{a}$

0celo7

like in high school algebra, $$\frac{1}{\sqrt 2-\sqrt 3}$$ was not an acceptable answer

Meow Mix

to get a difference of squares

0celo7

@MeowMix so do that here

Eric Silva

@Meow yeah it's what i suggested

Meow Mix

ok ok

$\frac{1}{2}a^{-1/2}$, just what calculus would have given me

0celo7

01:51

@AkivaWeinberger I need some of Balarka's heroin to figure this out

Akiva Weinberger

Heh :P

Figuring out the genus of your thingy, you mean?

Meow Mix

also if i have a function $f$ continuous at $a$, does $\lim_{x\to a} f(x) = f(a)$?

Akiva Weinberger

I actually have a few images saved on my phone that show how the deformation goes

Meow Mix

or do additional conditions need to be met

Akiva Weinberger

(or one way, anyway)

So coincidental that your icon happens to be a (smooth version of) that same surface

0celo7

01:53

@AkivaWeinberger yeah

so this is definitely a torus with a disk removed

Akiva Weinberger

Right.

0celo7

God I hate algebraic topology.

Akiva Weinberger

So it makes sense that your thing, which is like one step up, would be genus 2.

Meow Mix

looks like a tennis ball

0celo7

@AkivaWeinberger Yeah. It's a minimal surface btw

Akiva Weinberger

01:55

One thing you can do, other than visualize a deformation, is try to visualize the homotopy equivalence to 2g loops

Eric Silva

does this surface minimize something

gg sniped

Akiva Weinberger

@0celo7 Ahh

(cont'd) (which is slightly easier)

0celo7

Minimal surfaces are all ridiculous

Akiva Weinberger

2g loops wedge summed, I mean

Eric Silva

minimal surfaces are cool af

0celo7

01:56

It's such a strange condition

They look crazy

Eric Silva

i need to read colding and minicozzi asap

Akiva Weinberger

So your start with the boundary shape and see what soap film it would make?

Where did the boundary shape come from? @0celo7

0celo7

@EricSilva It's not that good

It goes from really technical to really sketchy

Pacing is off

Not a fan

Eric Silva

ive heard good things from the geometers at my school

Andre Neves seems to really like it

0celo7

I've heard good things too, but many people who know the stuff already tend to think everything is good

Eric Silva

01:59

that's maybe a good point

0celo7

@AkivaWeinberger I think the soap film stuff is something they tell people for grant money

Akiva Weinberger

Yeah, if you already know a topic (even somewhat) it's easier to forgive a book for being a little unclear

Transcript for

$Mathematics$ Mathematics