That's what you want for your proof. I ignored that sloppiness.

Yeah.

Thanks for your help :)

I didn't do much. :)

This problem is of the IMO.

Ah, interesting.

Now I'm not surprised.

Why?

I thought this was a bit sneaky for a standard calculus problem :P

Honestly, I thought it was trivial when I saw it.

Good night :)

@MikeMiller The statement of Reeb stability uses a holonomy assumption. Is there some holonomy assumption implicit in the compactness assumption?

im back!!

not that it's notable to any of you

but i am

Help.

Strike of the answerer who only answers one of the things I'm asking about and isn't even all that clear about which one he's answering.

Can anyone help me with a general polar equation? I'm given a circle with polar coordinates, say r = xsin(theta) + ycos(theta). There is a region shaded on the circle from [0,2]. If we revolve the region about the y-axis, what would be the volume of the solid generated? With the disk/washer method.

It's not so much about the answer, but rather the process that I

oops, hit send before finished typing

*that I am confused about

@Jessy regarding your confusing of multiplication

we're not multiplying two elements of $G$

We're multiplying $r_kg_k$ (an element of $\Bbb Z(G)$) with $r_ig_i$ (another element of $\Bbb Z(G)$).

btw i'm not really an algebraist (i'm stoopid), $\Bbb Z(G)$ is the field extension of $\Bbb Z$?

sorry this is a bit late, i was at the mall :/

and i didn't find this response worthy of an answer

@PVAL The version I mean is that a foliated neighborhood is determined by the germinal holonomy of the leaf (and indeed can just be written in terms of the holonomy). The compactness assumption implies the image of the holonomy in germs of homeomorphisms of (R,0) can only be the identity and negation.

Otherwise leaves will spiral forever.

@PVAL Well Mike set that as an exercise so he probably already knows.

On second thought I don't understand PVAL's objection anymore. A small, open, tubular neighborhood - once made transverse to the leaves - is a neighborhood of the leaf which is a union of the nearby leaves, not?

Hey everyone!

I don't remember why we cared about the boundary of the tubular neighborhood in the first place

@MikeMiller Do you need to know that once you understand there's a normal neighborhood foliated by nearby leaves? I mean then it's isomorphic to either the trivial line bundle on L foliated obviously or the "twisted" line bundle on L foliated by small S^0 bundles.

@AkivaWeinberger is it the case that countable sets are null?

... which is, I guess, exactly what you said.

@Zach Lebesgue measure zero?

hi @daminark

yeah

They are

The way to see it is this

Enumerate your countable set $\{a_1, a_2, \ldots\}$

ya

@ZachHauk Yes.

then you have a union of points

whos measures are all 0

That's one way to do it

Countable union is additive wrt measure

That would need to be proven ^

Now, this would require showing that Lebesgue measure satisfies the axioms of a measure

The other way is to cover $a_n$ with an open set of size $\epsilon/2^n$ and then you can finish from here I think @ZachHauk

also aren't finite unions of disjoint sets have the same measure as the added measure of each of those sets

@MikeM Anyway, the reason I asked this initially is because that proves your exercise. A small transversal is then hit by leaves in antipodes, more or less. That's the Z/2 action by reflection and gives it to you as a bundle over the orbifold over that.

Or otherwise, of course, the trivial bundle

what's a bundle

something ducks walk on

You can do it directly by taking intervals around each point whose total length sums to 1, then multiplying the whole thing by some $\epsilon$

@Akiva i haven't done anything real analysis-y

Hah @Zack

@ZachHauk It's the same argument as in the 3Blue1Brown video

right now i have to get done with ted's projective geometry stuff

then study algebra

to which extend idk

why

Extent?

idk i'm interested in algebraic geometry so that's what he told me to study

@Akiva pls

i'm not writing an essay

meh you should learn topology

I'm only saying that because it legitimately took me a few seconds to figure out what you were trying to say @ZachHauk

@Balarka i've done a bit of topology. plus Ted would get mad /s

all i've done is boring definition stuff

At some point you should learn the definition of compact

oh yeah, Ted would get mad. but what's life without adventure?

like "a basis is a collection of sets whose f inite unions makes up a topology blah blah"

Why would Ted get mad?

if all you've done is boring definitions you haven't really done topology :)

aka the good stuff

afaik topology is just definitions

that's all i've learned

hausdorff and stuff

No, it's not.

It all has meaning to it

as far as i know

not saying that's all there is :]

(and i know there's more because i haven't read the whole book yet!)

algebraic topology looks cool

What were you reading?

Munkres' part 1

Lol my topology knowledge comes from the appendix of Intro to Manifolds by Loring Tu

Well, I recommend studying metric spaces first. Much more fun.

Oh yeah Intro to Manifolds is a good book

^

Metric spaces are neat

@Bram28 Darn it. Every time I answer something I find that someone has sniped me

Plus my 3rd analysis pset

Noah Schweber is neat

Where we did 20 problems from chapter 2 of Rudin (that was an experience...)

Noah Schweber teaching metric spaces was neat

@BalarkaSen i've read a bit of

what's it called

But yeah I'd say you definitely want to get some amount of topology to navigate around

rudin's?

Rudin is a bad textbook. It's an exercise book.

I'm partway through Rudin's analysis text

A superb exercise book.

yeah that's the one i'm talking about

Hi, Brody

You should learn from Simmons.

@Akiva Little, Big, Grandpa, or Great Grandpa? :P

analysis

@Daminark Little

@Balarka i don't have the dough to purchase all these fancy books!

There's more than two?? @Daminark

all i have is the guts to pirate them (hush)!

who wants to buy books when you have them in libgen!

Well, little Rudin is the one you know, then there's Real and Complex Analysis

Then Functional Analysis

Then Fourier analysis on groups

Hello @Akiva and the rest

@Brody oh i see how it is

Just popping in for a sec

@Akiva is all great but you don't care about us???

@Daminark I've actually heard it called "Baby Rudin"

i feel attacked

Yeah, I alternate between "little" and "baby"

let us stage a rebellion

You didn't say hi to him :P

power to the proletariat!

DogAteMy personally greeted me in explicit terms

Zach, I'm pretty sure you're not in the proletariat?

Actually someone in my class made a condensed list of definitions and theorems which he called "Baby Baby Rudin"

That sounds great @Daminark

Do you have a link, actually?

But yeah Rudin's alright in some ways

Though you're all equally cool

@Akiva is working at an ice cream place considered the proletariat?

I stand corrected. Probably.

I would prefer if he gave a more integrated treatment of the stuff in chapters 2-4

Like, emphasize more that topology, continuity, and convergence are all secretly the same thing

Rudin not good? That's a first I've seen from this chat

Is your possession of highest value your ability to work?

@ZachHauk

i mean

Also chapter 7 somehow

Chapters 9-10 are u t t e r g a r b a g e

this keyboard is pretty valuable

@Daminark I don't think I'm up to those chapters yet

Dude when you get to multivariable calculus just don't use those chapters to learn from

They're a good reference

But like just don't try to learn from them

I have heard Ted criticize them

I'm in the middle of chapter 7

Chapter 7 is quality

my math is a mess right now

What's 7 about?

i don't even know where to go

@Zach Set it straight

Uniform convergence

Sequences and Series of Functions

Ah

Ends in the Stone-Weierstrass Theorem

all i have is Non-euclidean geometries which im almost done with

Chapters 5 and 6 are decent

In fact, I think I just finished chapter 7

and then, algebra

Chapter 11 is a bit iffy

I never really did chapter 8

but i don't know what algebra i need to know. like, group theory? ring theory? theory theory??!?!?!

I mean we didn't use just Rudin in my class

By favorite proof of Stone-Weierstrass is through approximations to identity.

There's a review here which I haven't read yet

What are the awful ninth and tenth chapters @Daminark?

(Fourier analytic in nature)

i guess Teddy- I mean Ted, will tell me what to do

@BalarkaSen I don't think I know that one

Actually, I could look that up

@Akiva Let me see if I have a link

The above link has a table of contents @Brody

I haven't touched chapter 10, but I've heard that it's just a lot of notation and you can't learn anything

Teddy Shifrin

I sat in on analysis last year for a bit

... O_O

@ZachHauk

???

Because originally I tried reading Rudin directly while I was just beginning Spivak

Which was brutal

what's so shocking

"Teddy"

idk people called Ted are called Teddy

like

The analysis class here does chapters 1-4 first quarter, chapters 5, 9, 6, and 7 second quarter (I started auditing in the winter)

So it basically goes downhill moving into multiple variables

roosevelt

@Brody I guess they sacrificed quality for more dimensions

opposite :/

But like, the multi part of that class is just the "Let $h = \frac{\epsilon}{34.7\sqrt{n}}$" sort of nonsense

i need to read spivak's too

And it seemed like he was reading off Rudin directly

but i also need to read artin;s

ahhhh so many things to readdddd

Don't worry @Zach, you got a ton of time

i'll never beat @Balarka at this rate /s

@ZachHauk Didn't even notice after having read it lel

@Akiva .pdf's are not opening in this end (slow internet) but the exposition is by Matthew Bond, "Convolutions and The Weierstrass Approximation Theorem"

Oh @Balarka so my 207 prof did that with us

Found it

And went into more detail on the convolutions

6 pages?

Yes, it's very short

We did some convolutions with Gaussian kernels

Spivak is nice but the language and exposition are for the less mathematically mature

And by some witchcraft I can't recall we arrived at the heat equation

@Zach If you truly love math your final goal shouldn't be to beat anybody at anything

Oh wait, speaking of Calculus I mean

@Balarka it was a lighthearted joke