Mathematics - 2019-12-03 (page 1 of 4)

« first day (3408 days earlier) ← previous day next day → last day (1614 days later) »

12:30 AM

Hello people

The lying over theorem says that A is a subring of B, with B integral over A, then every prime ideal of A is contraction of some prime ideal in B.

Is a more general statement true? If $f : A \to B$ is an integral homomorphism, then, is every prime ideal of $A$ contraction (under $f$) of some prime ideal of $B$?

$f$ being integral homomorphism means that $B$ is integral over $f(A)$.

More precisely, if $\mathfrak{p}$ is a prime ideal of $A$, then is $\mathfrak{p} = \mathfrak{q}^c = f^{-1}(\mathfrak{q})$, where $\mathfrak{q}$ is some prime ideal of $B$?

12:57 AM

well $f^{-1}(\mathfrak{q})$ is always a prime ideal of $A$ at least

lol

This Materials Modeling stack exchange is currently ranked number 1 in Area51 in terms of its progress: area51.stackexchange.com/proposals/122958/materials-modeling…. Let's support our neighbours that are using mathematics to model materials!

1:12 AM

@ÍgjøgnumMeg yeah. But we're asking about the converse. Is every prime ideal of $A$ is of that form.

The lying over theorem (See Theorem 5.10 in A&M) states that this is true for the special case when $f$ is the inclusion map, $A \hookrightarrow B$.

@ÍgjøgnumMeg bist du hier?

Ja

@ÍgjøgnumMeg ich habe mein Prof gefraget

uber $x^p - px^{p-1} + (np+1)p$

Was hat er gesagt? :D

he found a paper that linked to another paper

and that's the end of the story

1:21 AM

np^2 + 1 oder?

it's (np+1)p, I messed up

Ah nice :)

not that it makes any difference :P

so it's a mystery now

and maybe you can publish a paper if you figure out why it's galois :P

weird

steals

Howdy, @ÍgjøgnumMeg and @Leaky

1:23 AM

Hiya @Ted :)

@TedShifrin bonsoir

@ÍgjøgnumMeg have you learnt Dirichlet's theorem of prime?

Going through some elementary observations on the $\Gamma$ function

primes in arithmetic progression?

I saw a proof on it in the course on $L$-functions that I thought was very cool

what's it?

I can't quite remember the full proof but it essentially boiled down to showing that a certain divergent sum would converge if only finitely primes of the relevant form existed

which I thought was very nice lol

(as in, the sum was known to be divergent but could be shown to be convergent under the assumption that only finitely many primes of the form $a + kd$ existed, or something equally handwavy)

@AkivaWeinberger But you should listen to God's talk

"What is Manifolds?"

Akiva Weinberger

1:32 AM

Whose?

Gromov :3

@BalarkaSen wow you're here also

I am always here

Greetings, a @Balarka

@BalarkaSen wanna play?

1:36 AM

Hi @Ted!

Akiva Weinberger

Hi a Ted

@LeakyNun Alright. Not in a great shape, but I can try

Send me a link

Hey there, old man DogAteMy!

@BalarkaSen lichess.org/kCbxOuro

@BalarkaSen there must be a bug

Yeah

1:37 AM

because some other anonymous player also can't start with white

Some connection problem

so I must be white

Maybe I'll sign in and send you a friend req

ok sure

What was your username again lol

1:42 AM

@BalarkaSen ChEsSn0oBz

Of course :P

Makes perfect sense.

Shouldn't have started with Scandinavian, I don't know the first thing about it lmao

Scandinavian is a language?

I thought Scandinavia was a region ....

An opening rather

1:44 AM

@TedShifrin how far do you usually get in G&P in a semester (4 months)?

LOL, oh.

I did pretty much the whole book, but I did certain things differently, like the Poincaré-Hopf Theorem. I also covered Mayer-Vietoris for deRham cohomology and did examples.

I was writing out notes for this class on it and I realized most of the substance occurs in the exercises and explanations. The actual number of theorems isn't actually as many as I have seen in other courses.

I thought 200 pages warranted 2 semesters, but writing out just statements, it definitely seems like 1 semester as you teach it.

I wrote a number of extra problems, too (some more advanced for grad students), because I felt that Guillemin gave away too many hints. The one problem he needed to give hints was the one constructing path lifting of maps $S^1\to S^1$.

I also did the explicit example of the Hopf map $S^3\to S^2$ every time and showed them that fibers linked.

$\pi_1(S^1)$ is surprisingly hard to calculate.

Well, it was ridiculous that on that one he didn't give any hints.

Did I send you my G&P homeworks, a @Balarka?

Years ago, of course.

1:49 AM

Yup

I remember the problem on parallelizability of products of spheres.

@BalarkaSen did you read G&P?

Thought so. Just haven't made you do all the complex geometry exercises yet :P

Could someone please help me figure out this one

0

Q: Estimating number of people who selected a specific shape

Let's say we have 4 shapes (triangle, square, rectangle and circle) and 50 people are each asked to pick 1 of the four shapes. Based on statistics, it is said that half of the people (25) will select a triangle and the number of people who select a rectangle is twice the number who choose square....

@anakhro Yeah

@BalarkaSen I guess I am late to the game then. :P

1:51 AM

@DemCodeLines: Is this really statistics, or is it a basic algebra problem? I can't tell.

Never late to read a good book

I also gave you the exercise that a $C^k$ retract is always a $C^k$ submanifold when $k\ge 1$. That is still a cool exercise.

@TedShifrin It's from a Data Science problem set, which is why I labeled it as Stats

I know absolutely no statistics, @DemCodeLines.

@TedShifrin Yeah, constant rank theorem is an A+ theorem

1:52 AM

I originally solved it without using that theorem. Just from the submersion theorem.

I like that he doesn't opt for the modern notation and so you actually get to prove things rather than just learn a bunch of notation.

$SO(2)$ everyone here

Shout out to everyone here

@DemCodeLines: So there are approximations, obviously. If we chase through the facts, we have 25 triangles, 15 rectangles, 5 circles, 5 squares, adding up to 50. But we don't have 15=2(5).

So you're supposed to do some sort of data analysis. I dunno what.

@anakhro: I'm a big believer that one should learn diff top (including transversality) and just do submanifolds of $\Bbb R^n$, not worrying about all the abstract chart/diff structure stuff until later.

I tried using simple algebra, but lost it on 2(7.5) = 15, which makes no sense

Are questions related to the Black-Scholes model on topic for math stack exchange?

1:56 AM

Right, @DemCodeLines. Leave that one for last and you get what I got. But you're presumably supposed to do some statistical stuff to make a meaningful answer. That's beyond me.

@Bob: For the main site, sure. Mathematical finance is becoming reasonably common. It's just that we (most of us?) don't know any of it.

@Bob I think Economics S.E. would be better don't you think?

@TedShifrin do you have any hidden gems of books that do things with the modern notation (that you'd suggest after G&P)?

I never really liked either of the Lees' books, and Tu is okay, but kind of slow. I think I will probably pick up Bott & Tu next semester for fun.

No, @anakhro. I like Boothby, Spivak, and the world loves Lee (his books seem too wordy to me, but I don't know them well).

I tired quantitative finance a while back but I did not get a good answer

I taught my own graduate courses doing an amalgamation of my own thing (more with differential forms and vector bundles, because of my complex geometry bias), but certainly followed Boothby and Spivak in places.

2:00 AM

I like Spivak, but I am surprised you find Lee wordy but like Spivak.

I suppose Spivak tries to use his words to convey something, whereas Lee is variable.

thanks and good night

Guten nacht Bob

Yeah, certainly, over 5 volumes, Spivak is very wordy.

Is there a particular volume you liked?

I have used 1, 2, 3, and 5 the most.

2:03 AM

Matsumura was fun, but Federer was like playing tennis on a sunny day

For things like Gauss Bonnet, I prefer to teach characteristic classes the way Chern developed them, and do stuff with Schubert cycles in Grassmannians ... rather than the "modern" approach.

Is 4 the Riemannian one?

3 is definitely Riemannian. 4 does lots of PDE stuff.

Oh I see.

Does he ever do symplectic structures in any volume? I know his physics book does.

Arnold's PDE book is very good and even does the relevant contact stuff.

No, he discusses that (along with complex geometry, etc.) at the end of Volume 5 where he talks about various avenues of future exploration.

2:11 AM

For complex geometry do you have suggestions for favourite literature? Or do you prefer your own notes for that?

@Leaky what's the link for the paper you were talking about?

@anakhro: Chern's little book, Ronnie Wells's complex manifolds book, and, of course, Griffiths/Harris. People nowadays like Huybrechts and deMailly.

Did you ever consider writing a textbook on complex geometry, or were you satisfied with the notes for your classes?

@ÍgjøgnumMeg sciencedirect.com/science/article/pii/… 2.3.1

I should look into Arnold's PDE once

2:28 AM

@Leaky doesn't that theorem show that $K^g/\Bbb Q_p$ is Galois? lol

Oh sorry

that's the Galois closure

rofl

lol

#beingabletoread

Can somebody help me how to solve frobenius method

Frobenius might be able to help.

turns to the grave of Frobenius.

0

Q: Solving using Frobenius method

$2x^2y''-xy'+(x-5)y=0$ I know how to solve using power series but I am not able to understand the Frobenius method. Can somebody do the solution in a basic manner so that I can understand how to solve this.

ordinary-differential-equations

2:42 AM

@BalarkaSen wow SF says Qc3 instead of Rf7

Oh fuck

Nice

Didn't see that at all

@BalarkaSen I still don't see why

But see, it was winning for you because you had more queen side pawns

@LeakyNun I think queen checks, you move your king, then Rf7

then what

you have to take the rook with the queen otherwise it's mate

2:46 AM

it isn't

24. Qc3+ Kh6 25. Re3 g4 26. Rf7 Qg5 27. h3 Qh4 28. Qd4 Rh8 29. Kg2 c5 30. Qg7+

Re3? You mean Qe3?

I just copied the line

oh wait you wanted Rf7

Yeah

ah ok then it's R v Q

Yup

2:50 AM

cool

This stuff is too hard

chess is hard

How do people even play man

I would love to play again, but I have assignments

I have to sleep!

2:50 AM

you should

Also I think I'm done for today lmao

Good games, both

You play really well

gg

3:04 AM

My phone is charging but somehow the batter life is going down

It was like at 10% and now that I put it to charge it shows it's at 1%???

wtf

turn the cable around

or hang the phone under the laptop so the electricity flows down under gravity

#PhySiCs

Lmao my phone is charging the charging point

you're putting energy into the grid man

Altruist

so that's how they steal electricity

when you think you're charging your device, no, it's a government conspiracy

your device is charging the electricity supply

3:08 AM

There's a big vat of electricity sitting underground in the capital

just filling up with electricity

stolen from people using their charge cable the wrong way

this is how electricity works right

its capitalism bro

we're powering the computers in wallstreet

Hmm so I'm thinking about why $\Gamma(z)$ is holomorphic on the open right half plane

by defining $f_n(z) := \int_{\frac{1}{n}}^n t^{z-1}e^{-t}dt$

showing these guys converge on the closures of open balls in the right half plane I guess

$f_n \to f$ obviously

yeah, uniform limit of hol. functions on compact domains is hol.

and the $f_n$ are holomorphic obviously

so I just need to show that $f_n \to f$ uniformly on those closed balls

sounds right

3:12 AM

$f - f_n$ is just a tail of the Gamma function which I've just shown converges absolutely on all of the right half plane

I think?

actually its $\int_0^{\frac{1}{n}} +$ tail of the Gamma function

but that guy on the left is gonna go to 0 as $n\to \infty$

how barbaric

to use $z$

rofl

you'd prefer $s = \sigma + i\tau$

no!

it's $s = \sigma + it$

I think Riemann used $\sigma + i\tau$ and Riemann is bae

$\Gamma(k)$ with $k = \Sigma + j\Xi$

$\displaystyle \left| \int_0^\infty t^{s-1} e^{-t} \ \mathrm dt \right| \le \int_0^\infty t^{\sigma-1} e^{-t} \ \mathrm dt \le \int_0^\infty t^{N-1} e^{-t} \ \mathrm dt = (N-1)!$ for $1 \le \sigma < N$

this achieves nothing wait

3:18 AM

@ÍgjøgnumMeg Yeah it's just Weierstrass' M-test. If each term of a sum or integral is bounded then the sum or integral converges uniformly.

Nise one

oh this achieves something

this should tell you that the tails are uniformly vanishing lol

maybe not

but the idea is that you can bound it by one integral

one integral to rule them all

Yeah, your inequality, but just applied to the tail integrals instead

$\int \infty$

yeah

the fact that $\int_0^\infty t^{N-1} e^{-t} \ \mathrm dt$ is finite means that the tails go to 0

so yeah that's how you prove it @ÍgjøgnumMeg

3:20 AM

The holomorphy?

yeah

nise

the Morphy

The hol got-damn morphy

I got a fun exercise to do tomorrow

cool

3:22 AM

Prove that the Poincaré series of fixed weight form a basis for the cusp forms of that weight

@Leaky what was Morphy's defense again

just Ruy Lopez?

@BalarkaSen according to wiki it's 3... a6 in Ruy Lopez

Ah

but I don't play the Ruy Lopez lol

most people just take the knight after that lol

messy pawn structure

3:24 AM

> In the Exchange Variation, 4.Bxc6, (ECO C68–C69) White damages Black's pawn structure, giving him a ready-made long-term plan of playing d4 ...exd4 Qxd4, followed by exchanging all the pieces and winning the pure pawn ending. Max Euwe gives the pure pawn ending in this position (with all pieces except kings removed) as a win for White.[8] Black gains good compensation, however, in the form of the bishop pair, and the variation is not considered White's most ambitious, though former world champions Emanuel Lasker and Bobby Fischer employed it with success

yeah

I should read some book or something once in a while

same

I wanted to read Nimzowitsch's "My System"

but my blind chess playing friend said its trash lol

I tried to read it but maybe it wasn't very suitable for me

"Nigel Short has claimed that 'My System' should be banned."

lol

3:28 AM

lmao

talking about Short have you watched his famous game

the 7 games against "Bobby"?

the king march

yeah crazy

the opponent was like "what are you doing with your king"

and when he found out, it was too late

lmao

that had to be some engine dude tho

3:30 AM

what's more interesting is that engines have a hard time recognizing this

I tried once, it took some time before announcing Kh2 as the best move with eval +5

and then once Kh2 is played, the eval goes up to +15

Huh

4:00 AM

I really enjoyed that Elder album you suggested, thanks @BalarkaSen

I really enjoyed that Elder album you suggested, thanks @BalarkaSen

Any other recommendations?

@TedE Like Elder specifically, or something else?

They have a new EP which I also like

"Gold and Silver Sessions"

Very different, stylistically

Anything else, what are your favourite bands (of that style, or any other)?

Hmm

I am into a lot of loud stuff so I am not sure how you'll like them since you're more into soft/progressive stuff

What is your favorite Opeth album? :P

I can lead you somewhere from there maybe

HeRiTaGe

YES

4:15 AM

yaaas

Remarkable album I don't know why people don't like it as much

Not a clue, I love it

Heritage, Watershed, Blackwater Par

k

I can't tell you what I like the best but if I were to list down what I listen to most frequently then

Damnation, Blackwater Park and Still Life

Ah Ending credits is one of my favourite songs to play

rofl

It's great

I also like In My Time Of Need

4:20 AM

instagram.com/p/BTuWeOGhsH6

#Plug

I haven't actually listened to Heritage :O. I like 'My arms, your hearse', and blackwater park most probably

wow 134 weeks ago

@TedE Ok so you actually do like heavier stuff

Cool @ÍgjøgnumMeg

Some of it for sure yeah

I can reco you some atmospheric black metal

4:22 AM

Sounds good :D

Eldamar's cover of Land of the Dead by Summoning

Ulver "Bergtatt", Agollach "The Mantle", Drudkh "Autumn Aurora"

My top 3 albums

Bergtatt is a masterpiece

Eldamar is great

Do you know Winterfylleth?

4:24 AM

I'll check all these out, thanks :D. I've actually heard Agalloch's Marrow of the Spirit and enjoyed it

Nope, checking it out

wait

Check out

Led Astray in the Forest Dark

by Winterfylleth

You will know this song hehe

@ÍgjøgnumMeg Isn't that like the title of Chapter I in Bergtatt

it is indeed lol

Oh ok it is a cover

4:27 AM

the song is an English translation tribute by Winterfylleth lol

Ahh

4:39 AM

@ÍgjøgnumMeg This is pretty good!

@Balarka yeah, Green Cathedral is a great song, very long tho :P I saw these guys live in the UK and they were great

Sayan Chattopadhyay

You guys listen to scary music lol

Yeah we're followers of anti-Christ

TrveKvlt

We go about our day riding a bike at 120 km/h through mountain passes, burning Churches and listening to heavy metal

Sayan Chattopadhyay

4:41 AM

Hahahaha

Translation: Valfar gets caught in a snow storm on the way home from seeing his mum and fucking dies

lmao

black metal ist krieg bro

Nargaroth all the way

its no joke

I actually haven't listened to Nargaroth in ages

youtube.com/watch?v=vOLtRdSg6eE Trve German Black Metuhhhllll

4:44 AM

Seven Tears are Flowing To The River is my favorite song from that album itself

A fine song

Oh btw, I saw Mayhem live the other day

Oh damn

I gotta say

it was kinda lame

Gaahl's wyrd were much better

I haven't actually listened to too much Mayhem

I like the lore

some of their old songs are cool but most of it is boring music

4:45 AM

you know Necrobutcher said he wanted to kill Euronymous lol

rofl

I was on my way to kill Euronymous too

Lmfao

Did you watch Lords of Chaos?

Nope

Is it any good

it's quite good

the actor who plays Varg is lame

4:46 AM

I shall check it out

but the film itself is great

nobody can play Varg I mean come on

Louisss Cacheeet

yeah he's a character lol

Oh man I have become very fast at computing class groups for quadratic fields

at least for small square free integers

rofl

what's the class group of Q(sqrt(147))

shiet

se

c

4:50 AM

heh its not trivial according to

oeis.org/A003172

lol lemme just check that all the primes of $\Bbb Z[\sqrt{147}]$ lying over primes less than 13

ha nice

2 ramifies, 3 ramifies, 7 ramifies

How do you actually compute the class number in practice?

5 is inert

minkowski bound or something

oo 11 splits

fuck

4:53 AM

damn

@Ted you compute the Minkowski bound, and then you only need to check the orders of ideal classes of prime ideals lying over primes less than the Minkowski bound

this is kinda mechanical and easy for quadratic fields but often relies on finding solutions to Pell-like equations

which can be a pain in the ass

I tricked you actually

although there are algorithms to solve those so

147 is 3 * 7^2

oh fuk

4:56 AM

sry

okay so actually I'm computing the ideal class group of $\Bbb Z[\sqrt{21}]$

nah wait, 21 is 1 mod 4

damn boi u hurt me

lol rip

actually that makes it easier

cuz.. the minkowski bound is like.. less than 3

Ah

just need to check that $a^2 - 21b^2 = 2$ has a solution

I guess

which it does not

4:58 AM

how so? since its 1 mod 4 the norm is not that right

I actually think you can still take that

« first day (3408 days earlier) ← previous day next day → last day (1614 days later) »

Transcript for

Dec '193

$Mathematics$ Mathematics

Associated with Math.SE; for both general discussion & math qu...

join this room
about this room

00:00

06:00

12:00

18:00

all times are UTC

$Mathematics$

site design / logo © 2024 Stack Exchange Inc; legal

mobile