Mathematics - 2018-07-05 (page 2 of 2)

Rudi_Birnbaum

22:00

Grothendieck: "An actual prime? You mean like 57?"

also seemingly had no interest in actual examples ...

Maximus

LOL

geocalc33

Say you have a network. Would it be cool to study how the nodes grow as you add edges to the graph in a systematic way?

Leaky Nun

no, it wouldn't

geocalc33

how about studying evolving networks

Leaky Nun

nope

geocalc33

22:10

I think you're just saying no

2

but I can't prove it

Daminark

22:29

Technically he said "nope" the second time instead of "no"

Ted Shifrin

22:44

"Nope" to Demonark :P

Leaky Nun

if f is not nilpotent, then A_f is not the zero ring, so it has a prime ideal, so A has a prime ideal not meeting f

is this a valid proof of "nilradical is intersection of every prime ideal"

@Ted

Ted Shifrin

Ask Mathein.

loch

that seems fine to me - of course you're still using zorn's lemma here for the existence of a prime ideal

Leaky Nun

sure, I'm not saying I don't need to use zorn's lemma

loch

yeah i just wanted to make that clear :p since i remember very clearly that you need it to prove intersection of prime = nilrad

Mike Miller

22:50

Good morning

Leaky Nun

hi

@loch might you like to... check a proof for me?

loch

konichiwa

is it long/hard

2

Leaky Nun

long

loch

somehow that didn't come out the way i wnated it to

anyway

uh sure

Balarka Sen

Lol

Leaky Nun

22:51

lmao

now it surely didn't

Ted Shifrin

goodnight, @MikeM.

Leaky Nun

bonus if you know where it comes from :P

loch

oh is this AM

Balarka Sen

it's 2018, we're nearly adults, and we're still laughing at dick jokes

Leaky Nun

bingo

Ted Shifrin

22:52

speak for yourself, a @Balarka

Fargle

@BalarkaSen speak for urself I'm 23

Ted Shifrin

hides from Fargle

Fargle

:c

Balarka Sen

vote yes/no: should I grab this opportunity to learn measure theory? lmao

the end-goal being "post a massive answer expanding on that comment"

Leaky Nun

yes

you should learn measure theory, regardless of anything

Mike Miller

22:57

@Ted I forgot that joke. Did I always say morning?

I like this joke again. Actually, it is my morning right now.

geocalc33

Like this if you like mathematics

Ted Shifrin

Yeah, you did, @MikeM, and I always said good night :P

Mike Miller

good joke

Ted Shifrin

I wouldn't call any humor in this room "good."

Mike Miller

@BalarkaSen Measure theory isn't exciting but you should use it as an opportunity to learn measure theoretic dynamics

Like your comment

Ted Shifrin

22:58

I don't like measure theory much, but integration theory is great.

Balarka Sen

basically what I'd do is go through Brin-Stuck along the geodesic that lands me to the proof of what I stated

starting at measure theoretic basics

then write it all down lol

Mike Miller

Only write down what isn't tedious; link the important basic lemmas instead of proving them

loch

@LeakyNun looks good to me

Leaky Nun

thanks

Mike Miller

I'm going back to bed for a bit

Ted Shifrin

23:00

Night.

Wait, I already said good night.

Daminark

Yeah I dunno GMT was a bit fun but if there's anything in that general circle that I think I'd like it'd be ergodic theory

Mike Miller

Heh

Balarka Sen

I'd end up writing something like this, this or this

Leaky Nun

171

Q: Your favourite application of the Baire Category Theorem

I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications that confirm its neatness and/or power. Here's the theorem (with proof) and two applications: ...

I think I need to learn this one day

(might as well be today)

Balarka Sen

My last three mega answers that I remember

Leaky Nun

23:02

@loch do you have AM?

loch

i have internet

so yes

mick

I assume $\sqrt[3](1 - z^3)$ was supposed to be $\sqrt[3]{1 - z^3}$. Please let me know if I'm wrong. — Ennar 12 hours ago

Leaky Nun

so what exactly do I need to do for exercise 1.27?

I don't see anything to show

Ted Shifrin

@Leaky: Given a countable subset $S$ of the plane, prove that there's a point $x$ whose distances from the points of $S$ are all distinct.

loch

probably just to understand it

Daminark

23:03

I actually have no idea what ergodic theory is good for. The proof of the ergodic theorem and the fact that ergodic measures are extermal were both fun but I'm not sure what people really care about it for

Leaky Nun

@loch interesting

@TedShifrin is this related to BCT?

mick

-1

Q: How to solve $ f\left(\sqrt[3]{1 - z^3}\,\right)^2 = 1 - f(z)^2 $?

How to Find analytic $f(z)$ such that $$ f\left(\sqrt[3]{1 - z^3}\,\right)^2 = 1 - f(z)^2 $$ Koenigs function can not be used here So I am stuck. How does the riemann surface look like ?

dynamical-systems functional-equations riemann-surfaces complex-dynamics

Leaky Nun

what if S is just one point?

woah

Ted Shifrin

I guess one point is fine, anyhow, even though you're being ....

If you have only one number, it is not equal to any others on your list.

Leaky Nun

oh right

Balarka Sen

23:06

Finite is no problem still

Ted Shifrin

Right.

Balarka Sen

"theres a world of difference between diet coke and coke diet" - Akiva Weinberger 2018

4

Ted Shifrin

We don't see too much of DogAteMy these days.

2

In what?

Leaky Nun

@TedShifrin for any s,t in S, the set of points equidistant to s and t is a xxxxing line, which is nowhere dense.

Ted Shifrin

Oh, well, say hi to him for me, a @Balarka.

Leaky Nun

23:14

S is countable, so SxS is countable

is that it?

Ted Shifrin

Is a line actually nowhere dense in the plane? I've forgotten the definition.

Leaky Nun

i mean, a straight line

Alessandro Codenotti

@TedShifrin It is

Ted Shifrin

Yeah, yeah. That wasn't my quibble.

Leaky Nun

definition: S is nowhere dense iff S-closure-interior = empty

Ted Shifrin

23:15

I would say empty interior.

Ah, ok.

I'd forgotten.

It seems dense at its own points.

Semiclassical

hi ted

Ted Shifrin

hi @Semiclassic

Leaky Nun

every space is dense in itself =)

Ted Shifrin

But even thinking of those points in the plane, not just in the line.

Leaky Nun

every open set can be refined to a smaller open set that avoids the line

@BalarkaSen (is this argument right?)

Ted Shifrin

23:18

not if it's a neighborhood of a point in the line

Balarka Sen

@LeakyNun This is not my problem broham

Daminark

What does density at a point mean?

Leaky Nun

@TedShifrin eh, I mean, a smaller open set, that doesn't need to be a neighbourhood

@Daminark nothing

what is meaningful is that the point is a limit point

Ted Shifrin

Munkres carefully avoids this terminology, I believe. I'm with him.

Alessandro Codenotti

Among the interesting applications of BCT are the fact that complete metric spaces without isolated points must be uncountable, the fact that a function between complete metric spaces is continuous on a $G_\delta$ subset if you're looking for cool stuff that follows from BCT

Ted Shifrin

23:21

There are also cool things like deciding about the continuity of a pointwise-convergent sequence of continuous functions.

Alessandro Codenotti

Also a couple of "small lemmas" in functional analysis such as Banach-Steinhaus and the open mapping theorem if you're looking for "more serious" applications

Daminark

What I've seen as equivalent definitions of nowhere dense is that the closure has empty interior, and that there is no open set in which the set is dense

Ted Shifrin

OK, Demonark, I'll buy that.

Daminark

Oh regarding BCT there was one problem I had in my analysis final first quarter that involved BCT which I didn't get, I just mumbled something about S-W and was so salty when I realized

Alessandro Codenotti

Stone-Weierstrass?

Leaky Nun

23:24

@loch

Daminark

Yeah

Alessandro Codenotti

With the BCT you can also construct a perfect subset of $\Bbb R$ containing no rationals, which is completely useless as far as I know, but kinda cool

Leaky Nun

is there anything BCT can't do

Ted Shifrin

Oh, Rudin has an exercise like that. I think I figured it out eventually with nothing heavy.

Fargle

disprove BCT

loch

23:25

BCT can't help me decide what to get for dinner

@LeakyNun uh what's your question lmao

Ted Shifrin

@loch, perhaps you haven't asked it nicely.

Leaky Nun

@loch is that a sheaf morphism i'm looking at

is AM implicitly introducing the spec map

loch

oh
yeah that's secretly $\mathcal{O}_Y \rightarrow \phi_*\mathcal{O}_X$
well only on the global sections, but same idea

Leaky Nun

nice

Alessandro Codenotti

@TedShifrin Basically the proof I have in mind reduces to showing that $\bigcup_{q\in\Bbb Q}C+q$, where $C$ is the usual Cantor set doesn't cover the whole of $\Bbb R$, this can be shown with a few basic facts about the Lebesgue measure instead of invoking the BCT

Leaky Nun

23:31

use BCT sempre

even when non ci hai bisogno

Alessandro Codenotti

either "non ne hai bisogno" (you don't need it) or "non ce n'è bisogno" (there's no need for it)

Leaky Nun

ne

can I say non hai bisgono di lei

@AlessandroCodenotti

Daminark

Okay I can't find the final, I thought someone wrote up the problems sent it on our old group chat but apparently not

In any event it was something like

Like if you have a smooth function on a compact interval such that at any point, one of its derivatives is 0, then show it's a polynomial

I give no warranty on the adjectives

Alessandro Codenotti

@LeakyNun that's gramatically correct (you don't need her), but "lei" would be understood as referring to a person most likely

Leaky Nun

46

A: Your favourite application of the Baire Category Theorem

If $P$ is an infinitely differentiable function such that for each $x$, there is an $n$ with $P^{(n)}(x)=0$, then $P$ is a polynomial. (Note $n$ depends on $x$.) See the discussion in Math Overflow.

is it BCT on n?

Alessandro Codenotti

23:42

Discrete spaces are Baire, but I don't think that's how it's being used in that answer

I'll check that MO thread tomorrow though, it's almost 2am here

Leaky Nun

no, I mean, for each n, take note of the set {x | P^(n)(x) \ne 0}

probably doesn't work though

never mind

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