Mathematics - 2018-02-15 (page 3 of 4)

Balarka Sen

21:00

i drew a circle once but i fucked up so much it had trivial homotopy groups in all dimensions (except 0) but was nevertheless not contractible

(bonus meme: what space did I end up drawing?)

Eric Silva

dude that's an exercise in hatcher

Balarka Sen

yeah

Eric Silva

comby boi is also an example: topospaces.subwiki.org/w/images/1/1c/Doublecombspace.png

Daminark

That's not a space, that's hot garbage

Balarka Sen

lmao

Eric Silva

21:03

garbage theory is really hot rn

Balarka Sen

i'd forgotten the double comb is not contractible

wilkersmon

any differential geometers here?

Balarka Sen

but of course it isn't

wilkersmon

perhaps someone could help me by answering this question?

0

Q: Geodesic equation on a codimension 1 submanifold of $\mathbb{R^{n+1}}$

Section 8.4.3 in Taubes' textbook on Differential Geometry, from which I'm self-studying, is as follows: What is that convenient coordinate chart that Taubes is referring to? Could someone provide me with a proof that the geodesic equation takes the form that the author describes?

or at least giving me some hints

Balarka Sen

if i try to swoonswoosh everything to the wedge point i get fuckducked

because one after another middle middle ping pong

QED

Daminark

21:05

Balarka if a student writes that on a pset I will give them a negative score

It'll be so negative they will fail the class even if everything else is perfect

Eric Silva

"convenient coordinates" frequently means normal coordinates

wilkersmon

not sure how that helps me as I know next to nothing about normal coordinates

Eric Silva

then read about normal coordinates

they're very simple and are convenient for expressing the geodesic equation

MatheinBoulomenos

@Daminark since there's always a need for different expressions that mean "nice" or "not utterly pathological", I think it would be great if some niceness condition was named "non-garbage"

Eric Silva

semi-garbage

or quasi-garbage is good too

MatheinBoulomenos

21:16

semi-locally quasi-garbage

Balarka Sen

locally trivial garbage bundle

Daminark

"This nice is not the same as the last nice" -Marianna

ALannister

Anybody here who can help me with ODEs?

1

Q: Using bounds for an analytically intractable differential equations problem

Exercise 2.8.6 from Strogatz's Nonlinear Dynamics and Chaos with Applications asks us to consider the initial value problem $\dot x = x+e^{-x}$, $x(0)=0$ $\left(\dot x\, \text{means}\, \frac{dx}{dt}\right)$, which cannot be solved analytically. Part (b) of the exercise states as follows: U...

differential-equations dynamical-systems initial-value-problems

Semiclassical

@ALannister just as a reference point, here's what Mathematica's NDSolve plots for that ODE:

ALannister

That's time versus x?

Semiclassical

21:26

right

so it's mostly linear plus a small correction

ALannister

Good to know. So, roungly speaking, x(1) = 1?

Or x(1) = 1.2.

Semiclassical

yeah, lemme grab the specific number

x(1)=1.15364

ALannister

Really, what I am having trouble with is figuring out what the intitial conditions are for my bounds.

Semiclassical

no idea how precise that really is

ALannister

I'm not even sure whether I want to use the IC's from the function I'm estimating

Semiclassical

21:28

well, what I notice from the graph is that it's mostly linear

ALannister

Basically, I want to bound it above and below by two easily integrable functions.

Semiclassical

which isn't so surprising: If we say $x\approx 0$ for small $t$, then $\dot{x}=x+e^{-x}\approx 1$

ALannister

Then I want to integrate each, and let t=1 in each of them so I can find my constants of int.

But, the problem is, when t=1 in each of them, I don't know what x is supposed to be

Semiclassical

yeah

I mean, my heuristic reasoning is that $x$ is increasing at $t=0$ since $\dot x=1$ then

so we expect $x>0$ for $t>0$

ALannister

I also posted on chegg, and the probvlem has actually been answered 3 times there, but incorredctly. They all used integrating factors, which is not what we're supposed to be doing.

Semiclassical

21:31

and then $\dot{x}>x$ since $e^{-x}>0$

so one 'should' end up with $x(t)\geq t$ as a lower bound

Akiva Weinberger

Japanese sentence order is also really weird. "先週に映画を見た人は誰？" is literally "last week ni movie o saw person wa who?" (and translates to "Who is the person who saw the movie last week?").

Semiclassical

For an upper bound, I want to say you can take $\dot{x}=x+e^{-x}\approx x+(1-x+x^2)=1+x^2$ which is solvable

Akiva Weinberger

(And by "weird" I mean "alien, foreign") (@ÍgjøgnumMeg @Slereah @Secret If you care ^)

ALannister

x(t)? I don't know. That Evgeny guy said $x \leq x + e^{-x} \leq 1 + x$

Semiclassical

(not that I remember how off the top of my head)

ALannister

21:33

I had $1+ x^{2}$ as the upper bound in what I posted.

Semiclassical

ah, nice

I don't know how to make this rigorous, tho

so I can't help there

ALannister

@Semiclassical you just solve $\frac{dx}{dt} = 1+x^{2}$. I already made the bound rigorous.

ÍgjøgnumMeg

@AkivaWeinberger That's cool, I know that Japanese uses a lot of particles to modify meaning in sentences, I'm assuming that's what the "ni, o, wa" are? Also, that is sort of similar to Turkish, which is one of the reasons Japanese and Turkish were "controversially" thought of as belonging to a language family called Altaic

ALannister

Acutally, @Semiclassical it's $\frac{dx}{dt} = 1 - \frac{x^{2}}{2}$.

ÍgjøgnumMeg

@AkivaWeinberger i guess "literally" it's something like "the last week movie seeing person is who?"

ALannister

21:37

And I got $\frac{1+\frac{x}{\sqrt{2}}}{1-\frac{x}{\sqrt{2}}}= Ce^{\sqrt{2}t}$

You don't even want to try explicitly solving that for $x$

For $\frac{dx}{dt} = x+e^{-x}$, I'm given that $x(0)=0$ but I'm not told anything about what happens at $x(1)$.

Semiclassical

Ah

@ALannister i think that's actually not as bad as you do, fwiw

ALannister

fwiw?

Semiclassical

for what it's worth

ALannister

Oh, kids these days and your internet acronyms ;P

@Semiclassical how so?

Semiclassical

adding 1 to both sides makes it $\frac{2}{1-x/\sqrt{2}}=1+Ce^{t\sqrt{2}}$

Akiva Weinberger

21:42

@ÍgjøgnumMeg More or less, "last week ni movie o saw" is a verb clause than translates more like "last week, saw movie". But you're using it as an adjective to describe "person."

Also, there are no plurals in Japanese, so those can be movies and people.

Semiclassical

And then some obvious rearrangements give $$x=\sqrt{2}\left(1-\frac{2}{1+Ce^{t\sqrt{2}}}\right)=\sqrt{2}\left(\frac{Ce^{t\‌sqrt{2}}-1}{Ce^{t\sqrt{2}}+1}$$

sighhh

Akiva Weinberger

And I'm not even sure it translates to "the movie", it's probably "a movie" actually

(Yeah, those are particles)

ALannister

Interesting.

ÍgjøgnumMeg

@AkivaWeinberger Yeah I think that's basically the same as in Turkish, I THINK (my turkish is poor, if there are any turkish speakers here please correct me) the sentence in Turkish is something like "Gecen hafta bir film izleyen adami kimdir?" which would translate literally to "Last week the a film watching man is who?" so something like "The man who was watching a film last week is who?"

ALannister

Your latex got screwed up again

Semiclassical

21:44

But $x(0)=0$, so $C=1$

Akiva Weinberger

That's cool. Yeah, I can definitely see why Altaic was thought to be a thing

Semiclassical

blah

my connection is being an arse right now

ALannister

But $C=1$ only for $x(0) =0$

ÍgjøgnumMeg

@AkivaWeinberger Verbs with "-en" on the end tend to be adjectival verbs if I remember correctly, turkish lacks relative clauses so a lot of sentences tend to be flipped compared to English and you see a lot of adjectival verb forms (I think my linguistic terminology is correct). Cool to see more people interested in linguistics!

Semiclassical

uh

ALannister

21:46

If I want x(1) = something, I need another C, don't I?

Semiclassical

C is a constant

ALannister

RRRR drrr.

Drrr drr

Drrrrrr

Semiclassical

I've had days like that

ALannister

You're right. It's not as bad as I thunk it. ;P

Ever since the end of November/December, every day has been like that for me.

Semiclassical

Though, I think there's something weird---when I ask mathematica to solve the ODE because i'm lazy, I get $\sqrt{2}\tan(z/\sqrt{2})$

whereas the answer above would be tanh

ALannister

21:47

Yeah, it's not tan.

It's tanh-1

Could it be because z is complex?

Semiclassical

no, shouldn't be. and I actually meant to say tan(t/sqrt(2))

ALannister

arctanh

Hmm...

Think I did something wrong?

I actually had Wolfram Alpha evaluate the integral, because I'm awesome like that

Semiclassical

I think so, but it shouldn't be anything major

probably just a sign thing

ALannister

But maybe I set it up wrong?

Semiclassical

yeah, recheck it to be careful

The ode I end up solving is $\dot{x}(t)=1+x(t)^2/2$ with $x(0)=0$

ALannister

21:50

Yeah, see the sign inside the second log?

Semiclassical

yeah. hmm

ALannister

So, I mos def made a mistake.

Semiclassical

kk

ALannister

Probably in how I applied the identity.

B/c Wolfram Alpha didn't explicitly do that

So, I'll have to go back and fix.

Semiclassical

right.

ALannister

21:52

But what you're saying is, apply the IC for x(0), and I should get the C that I want that will let me be able to solve for x

Akiva Weinberger

@ÍgjøgnumMeg Can I tell you something cool about Hebrew? (Since I know much more about Hebrew than Japanese - I'm gonna say that I'm "preparing myself to learn Japanese" rather than actually learning Japanese because even though I've gotten quite far into this online grammar textbook I'm barely putting in any effort to remember the vocabulary, and I can barely put together a sentence on my own)

and I actually know Hebrew

Semiclassical

right

Akiva Weinberger

So you've heard of prefixes and postfixes (like in English, -ed is a postfix for the past tense)

ÍgjøgnumMeg

@AkivaWeinberger Is it to do with the weird triliteral roots that semitic languages have? hahaha

ALannister

Okahy, here goes nothing. I'm gonna grab a glass of wine and try again...

Akiva Weinberger

21:53

In Hebrew, you have transfixes.

@ÍgjøgnumMeg Yeah

MatheinBoulomenos

I just got a mail asking me if I want to TA a number theory course next semester

sweet

ÍgjøgnumMeg

@MatheinBoulomenos Nice!

@AkivaWeinberger Yeah semitic languages are crazy, are you a native speaker or religious learner or something?

ALannister

Thanks @Semiclassical. Let $A=\{\text{number of times Semi has pulled my arse out of the fire}\}$. Then, $A$ is uncountably infinite.

Semiclassical

hmm, but when I do your integral I get $x(t)=\sqrt{2} \tanh(t/\sqrt{2})$

which agrees with what you had before

so now I'm confused.

wtf mathematica

ALannister

We should be getting the same thing, b/c it pretty much is wolfram alpha

Semiclassical

21:55

right

ALannister

Only thing is, I'm solving the indefinite integral

Semiclassical

yeah

Daminark

@MatheinBoulomenos Awesome!

Semiclassical

oh doy

ÍgjøgnumMeg

@MatheinBoulomenos Salaried position?

Semiclassical

21:56

$\frac{dx}{dt}=1+\frac{x^2}{2}\implies dt=\frac{dx}{1+x^2/2}$

MatheinBoulomenos

@ÍgjøgnumMeg yes

ÍgjøgnumMeg

@MatheinBoulomenos Congrats :)

ALannister

Right

MatheinBoulomenos

@ÍgjøgnumMeg thanks

Akiva Weinberger

@ÍgjøgnumMeg Sorry, I had to go for a minute

ÍgjøgnumMeg

21:57

@MatheinBoulomenos Cool that you TA something you're actually interested in, I TA basic calculus and it's boring af

Akiva Weinberger

@MatheinBoulomenos Yay

ALannister

Then, let $u = x/\sqrt{2}$

Semiclassical

So the denominator should be 1+x^2/2 not 1-x^2/2. Was that the sign you noticed?

Akiva Weinberger

@ÍgjøgnumMeg So the root for "talk" is dbr, which you can't pronounce

but the past ("he talked") is dabar.

ALannister

No, your sign is wrong.

Lies

My sign is wrong

Your sign is right.

I'm doing the wrong thing.

Argh

Akiva Weinberger

21:59

The past first person ("I talked") is dabarti. "She talked" is dabra. Present (male singular) is medaber

Semiclassical

Yeah. I missed it at first

That should end up turning tanh to tan

Akiva Weinberger

"They will talk" is yedabru

ALannister

I was wondering what kind of sadistic diff eqs book would give us an integral with inverse hyperbolic tans in it

ÍgjøgnumMeg

@AkivaWeinberger Crazy, is there any actual structure to the formation of verbs, or is it just... stick some letters around the triliteral and call that something?

Semiclassical

Ya

ALannister

22:00

Okay grins

Akiva Weinberger

So the morpheme for future third person plural, at least for this type of verb, is ye_a__u

ALannister

Still, this clears a whole lotta shhhh up

Semiclassical

Yeah

ALannister

Hopefully then, I can still solve for x easily?

what did you say you got for your antiderivative again?

Akiva Weinberger

@ÍgjøgnumMeg Yeah, there's seven types of verbs, and they all conjugate in different (but roughly similar) ways

Semiclassical

22:01

I didn’t say, actually

ÍgjøgnumMeg

@AkivaWeinberger Ridiculous, I am truly molly-coddled in my study of Germanic languages. hahaha

Daminark

Wait are there unsalaried TA positions? Who would take those?

Semiclassical

In this case I don’t know what the antiderivative should be

Akiva Weinberger

@ÍgjøgnumMeg Oh, typo, it's dibar and dibra, not dabar and dabra.

ALannister

All right lemme go try

Akiva Weinberger

22:02

Dabar would happen if it were one of the other types of verbs.

ÍgjøgnumMeg

@Daminark sometimes students can just take "helper-outer" style positions for experience or something, I think I know a couple of people who do that here

Akiva Weinberger

@ÍgjøgnumMeg Essentially, if you want to make something passive, you shove its three-letter root into one of the other seven templates

Daminark

Ah, that I can buy, I guess I usually associate TA with some kind of grading

Semiclassical

But the change amounts to replacing $x\mapsto ix$ and so the fact that MMA gave tan not tanh makes some sense

Daminark

And grading is the last thing I'd want to do for free

PVAL-inactive

22:03

for fuck all is a better description than experience

MatheinBoulomenos

yes, I have to grade as well

Semiclassical

Hopefully the antiderivative in this case is just arctan

And I think it is?

MatheinBoulomenos

but it's okay, I also get to teach and answer questions etc.

ÍgjøgnumMeg

@Daminark Ah right, I don't grade anything hahaha, I just teach on a foundation course for people who did shite in their A-Levels

Akiva Weinberger

So three templates for active verbs, three templates for passive verbs corresponding to the first three, and three templates for reflexive verbs. More or less. There are exceptions @ÍgjøgnumMeg

ÍgjøgnumMeg

22:05

@AkivaWeinberger That's crazy, I usually use Arabic as my quintessential "really hard grammar" language when people ask about that kind of thing, I only know a couple of roots like "s-l-m" for "peace and religion kinda junk" and "k-t-b" for "books and writing and education kinda junk"

Daminark

Yeah teaching/answering questions is a whole lot of fun

I graded for discrete and I do have some stories from there though

ALannister

Yup, it's $\sqrt{2} \arctan \left(\frac{x}{\sqrt{2}} \right) = t + C$

So, now, is there a way to explicitly solve for $x$?

Daminark

One homework assignment had a problem of finding the probability that a k-length word over an n-letter alphabet has no repeated letters

ÍgjøgnumMeg

@Daminark Definitely, though I have one student who won't shut up and always asks if I can prove Fermat's last theorem while I'm talking about like.. the power rule for derivatives

ALannister

I suppose I could divide through by $\sqrt{2}$ and then take tan of both sides

Akiva Weinberger

22:06

I think the passive form of diber, medaber, etc. is dubar, medubar, etc.

Daminark

Of course it's $n(n-1)\ldots (n-k+1)/n^k$

One of the answers I got was hilarious though

Akiva Weinberger

And the reason I know Hebrew is 'cause Judaism @ÍgjøgnumMeg

and I go to Israel fairly often

Daminark

$\dbinom{27}{k}/n^k$

Akiva Weinberger

Not fluent by a long shot, but whatever

ÍgjøgnumMeg

@AkivaWeinberger That's cool, a fine reason to learn a language

@Daminark lol

Daminark

22:07

Like, first off just the form of that answer is a bit wrong, but alright you forgot a $k!$ in the numerator

ALannister

But even without that, if I sub in the IC: $\sqrt{2}\arctan\left(0/\sqrt{2}\right) = 0 + C$, I get $C = 0$?

Daminark

$27$? Okay maybe you're thinking about the English alphabet plus another letter

MatheinBoulomenos

My favorite grading experience is the answer to the question "Are $A$ and $B$ isomorphic?". The answer was "$A$ is isomorphic. Proof: [Bunch of nonsense]. $B$ is not isomorphic. Proof: [More nonsense]"

3

Daminark

But wait in the denominator $n$ appears. Like, this person knew the length of the alphabet was $n$

Akiva Weinberger

The reason I wanted to learn Japanese is because Modern Hebrew has a lot of Indo-European influence, and Spanish (the other language I know) is very close to English. I wanted to see what something sufficiently foreign looks like

Daminark

22:09

@Mathein lmfao

ÍgjøgnumMeg

Hilarious

ALannister

@Semiclassical is that bit right? That $C = 0$?

Akiva Weinberger

Japanese doesn't even have legit pronouns, just twenty different nouns that mean "I" with different connotations (with regards to politeness, masculinity, etc.)

(Masculinity meaning macho-ness, I think, not grammatical gender, which Japanese doesn't have.)

PVAL-inactive

Talk about math not languages so I feel less dumb please.

ÍgjøgnumMeg

@AkivaWeinberger Faiiiiiiiiiir, I study everything fairly restricted to European languages, I'm learning Polish at the moment, but most of my language knowledge is from Germanic languages, specifically I speak English, Swedish, and German fluently, and I speak Norwegian and Faroese fairly well

MatheinBoulomenos

22:10

@AkivaWeinberger relevant movie scene youtube.com/watch?v=JIONqTpvqS4

ÍgjøgnumMeg

@PVAL-inactive Sorry, it's probably a bit obnoxious to be talking about it in a math-chat room

lol

Daminark

Uh, em... What's the 4th homology group of $\mathbb{R}^7$?

:thinking:

PVAL-inactive

don't take me seriously

Balarka Sen

:blobhyperthonkfast:

Akiva Weinberger

@PVAL-inactive New flair?

PVAL-inactive

22:12

@Daminark If you don't explain further I can only know the answer up to isomorphism class.

@Akiva Mine always changes randomly.

Akiva Weinberger

I'm gonna go soon, about to board a plane from Israel back to New York

ÍgjøgnumMeg

@AkivaWeinberger I heard a very cool story about written Chinese the other day. Since Chinese written language is for the most part independent from the spoken language, any two non-mutually intelligible languages using the Chinese system can understand each other's writing, because the writing system isn't based on constructing words from graphemes

@AkivaWeinberger So, in particular, some English guy goes over there and a Chinese man asks him to tell him what a piece of writing says, but the writing is in French, and the English guy had to explain to the Chinese man that he couldn't understand the French writing even though it's written with the same writing system as English

Daminark

@PVAL lmao

Semiclassical

@ALannister yeah

And then you can solve for x by rearranging to get arctan x and then inverting

Antonios-Alexandros Robotis

22:29

hi

MatheinBoulomenos

Hi @Antonios

Antonios-Alexandros Robotis

just finished all the lectures and such

MatheinBoulomenos

How was it?

Antonios-Alexandros Robotis

not too bad, was pretty informal haha

MatheinBoulomenos

This was the introduction to modules that we talked about?

Daminark

22:30

Hopefully your voice box didn't give out? Yesterday you seemed already like you had a sore throat and you had a lot to do today

Antonios-Alexandros Robotis

yes @MatheinBoulomenos

and my voice was fine

fortunately

sore throat now though lol :P

0celo7

@Semiclassical Should a talk title be short and sweet or descriptive

I don't have a natural title for this one

"The isoperimetric problem and the Sobolev inequality" works

Daminark

F

Semiclassical

22:46

Depends on the audience

0celo7

@Semiclassical the sole algebraist who was coming to the seminars stopped coming

Semiclassical

If you’re doing something like a survey, a light touch on the title makes sense

0celo7

I don't know how to make this sound more inviting

Semiclassical

Well, what’s it about?

0celo7

see the title above

MatheinBoulomenos

22:48

talk titles should be clickbaity "You won't believe these 5 amazing applications of the Sobolev inequality"

3

Antonios-Alexandros Robotis

lol

Semiclassical

:/

0celo7

number 4 is shocking

Antonios-Alexandros Robotis

Geometric Measure Theorists hate him!

3

0celo7

the talk is about GMT, but no one can know that until they're there and trapped

Semiclassical

22:49

3) it will keep your talk physicist-free!

0celo7

@Semiclassical When I start my black hole saga I will send the notices to the physics mailing list

Daminark

I don't know of too many people who wouldn't go to a GMT talk but who would go for Sobolev spaces

0celo7

basically most PDE people...

Daminark

Why are they averse to GMT?

0celo7

it's not good

Daminark

22:52

Isn't it likely to be useful to their stuff?

0celo7

nope

ALannister

What's a good lower bound to choose?

x?

0celo7

it's useful for geometry

and, in this case, a Sobolev inequality

Semiclassical

@ALannister I want to say dot-x is bounded below by its value at 0

ALannister

Which is 1

Semiclassical

22:54

So just dot-x>=1

Right

ALannister

It did say try being clever and whatnot

Tuki

Hello

Semiclassical

Heh

Balarka Sen

0

A: How to actually find a minimizing path on a manifold?

This answer is mostly an attempt at writing down a chunk of the theory of variations on the fly while learning it. The two major references this answer is based on are (1) Milnor, "Morse theory" (part III) (2) Oancea, "Morse theory, closed geodesics, and the homology of free loop spaces" (chapter...

I am tired

ALannister

Me too.

0celo7

22:56

@Semiclassical this has applications to capactitors

ALannister

I've been tired for 2 months. I'm already tired tomorrow.

0celo7

I should totally get some physics people

Semiclassical

Lol

ALannister

Olivia Newton John's #1 hit, "Let's Get Physics"

Tuki

Someone knows why integers are denoted with $\mathbb{Z}$ ?

Balarka Sen

22:57

Zahlen

ALannister

Something to do with the Germans

Semiclassical

I used a conformal map to solve a capacitor problem today

Tuki

Well is real numbers $\mathbb{R}$ as well related to germans somehow ?

Semiclassical

Was pretty pointless but I did like the answer

Tuki

Set theory invented by germans ?

Semiclassical

22:59

what I’m forgetting is Q for rationals

Transcript for

$Mathematics$ Mathematics