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

feynhat

00:30

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 $B$?

ÍgjøgnumMeg

00:57

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

lol

Senegal

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!

feynhat

01:12

@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, $A \hookrightarrow B$.

Leaky Nun

@ÍgjøgnumMeg bist du hier?

ÍgjøgnumMeg

Ja

Leaky Nun

@ÍgjøgnumMeg ich habe mein Prof gefraget

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

ÍgjøgnumMeg

Was hat er gesagt? :D

Leaky Nun

he found a paper that linked to another paper

and that's the end of the story

ÍgjøgnumMeg

01:21

np^2 + 1 oder?

Leaky Nun

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

ÍgjøgnumMeg

Ah nice :)

Leaky Nun

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

ÍgjøgnumMeg

weird

steals

Ted Shifrin

Howdy, @ÍgjøgnumMeg and @Leaky

ÍgjøgnumMeg

01:23

Hiya @Ted :)

Leaky Nun

@TedShifrin bonsoir

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

ÍgjøgnumMeg

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

Leaky Nun

what's it?

ÍgjøgnumMeg

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)

Balarka Sen

@AkivaWeinberger But you should listen to God's talk

"What is Manifolds?"

Akiva Weinberger

01:32

Whose?

Balarka Sen

Gromov :3

Leaky Nun

@BalarkaSen wow you're here also

Balarka Sen

I am always here

Ted Shifrin

Greetings, a @Balarka

Leaky Nun

@BalarkaSen wanna play?

Balarka Sen

01:36

Hi @Ted!

Akiva Weinberger

Hi a Ted

Balarka Sen

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

Send me a link

Ted Shifrin

Hey there, old man DogAteMy!

Leaky Nun

@BalarkaSen lichess.org/kCbxOuro

@BalarkaSen there must be a bug

Balarka Sen

Yeah

Leaky Nun

01:37

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

Balarka Sen

Some connection problem

Leaky Nun

so I must be white

Balarka Sen

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

Leaky Nun

ok sure

Balarka Sen

What was your username again lol

Leaky Nun

01:42

@BalarkaSen ChEsSn0oBz

Balarka Sen

Of course :P

Ted Shifrin

Makes perfect sense.

Balarka Sen

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

Ted Shifrin

Scandinavian is a language?

I thought Scandinavia was a region ....

Balarka Sen

An opening rather

anakhro

01:44

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

Ted Shifrin

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.

anakhro

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.

Ted Shifrin

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.

Balarka Sen

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

Ted Shifrin

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.

Balarka Sen

01:49

Yup

I remember the problem on parallelizability of products of spheres.

anakhro

@BalarkaSen did you read G&P?

Ted Shifrin

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

DemCodeLines

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....

statistics

Balarka Sen

@anakhro Yeah

anakhro

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

Ted Shifrin

01:51

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

Balarka Sen

Never late to read a good book

Ted Shifrin

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.

DemCodeLines

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

Ted Shifrin

I know absolutely no statistics, @DemCodeLines.

Balarka Sen

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

Ted Shifrin

01:52

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

anakhro

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.

Ultradark

$SO(2)$ everyone here

Shout out to everyone here

Ted Shifrin

@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.

DemCodeLines

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

Bob

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

Ted Shifrin

01:56

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.

Ultradark

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

anakhro

@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.

Ted Shifrin

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).

Bob

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

Ted Shifrin

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.

anakhro

02:00

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.

Bob

thanks and good night

Ultradark

Guten nacht Bob

Ted Shifrin

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

anakhro

Is there a particular volume you liked?

Ted Shifrin

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

Ultradark

02:03

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

Ted Shifrin

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.

anakhro

Is 4 the Riemannian one?

Ted Shifrin

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

anakhro

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.

Ted Shifrin

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

anakhro

02:11

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

ÍgjøgnumMeg

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

Ted Shifrin

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

anakhro

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

Leaky Nun

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

Balarka Sen

I should look into Arnold's PDE once

ÍgjøgnumMeg

02:28

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

Oh sorry

that's the Galois closure

rofl

Leaky Nun

lol

ÍgjøgnumMeg

#beingabletoread

Aladdin

Can somebody help me how to solve frobenius method

anakhro

Frobenius might be able to help.

turns to the grave of Frobenius.

Aladdin

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

Leaky Nun

02:42

@BalarkaSen wow SF says Qc3 instead of Rf7

Balarka Sen

Oh fuck

Nice

Didn't see that at all

Leaky Nun

@BalarkaSen I still don't see why

Balarka Sen

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

Leaky Nun

then what

Balarka Sen

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

Leaky Nun

02:46

it isn't

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

Balarka Sen

Re3? You mean Qe3?

Leaky Nun

I just copied the line

oh wait you wanted Rf7

Balarka Sen

Yeah

Leaky Nun

ah ok then it's R v Q

Balarka Sen

Yup

Leaky Nun

02:50

cool

Balarka Sen

This stuff is too hard

Leaky Nun

chess is hard

Balarka Sen

How do people even play man

Leaky Nun

I would love to play again, but I have assignments

Balarka Sen

I have to sleep!

Leaky Nun

02:50

you should

Balarka Sen

Also I think I'm done for today lmao

Good games, both

You play really well

Leaky Nun

gg

Balarka Sen

03:04

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%???

Leaky Nun

wtf

ÍgjøgnumMeg

turn the cable around

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

#PhySiCs

Balarka Sen

Lmao my phone is charging the charging point

ÍgjøgnumMeg

you're putting energy into the grid man

Altruist

Balarka Sen

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

ÍgjøgnumMeg

03:08

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

Balarka Sen

its capitalism bro

we're powering the computers in wallstreet

ÍgjøgnumMeg

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

Balarka Sen

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

ÍgjøgnumMeg

and the $f_n$ are holomorphic obviously

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

Balarka Sen

sounds right

ÍgjøgnumMeg

03:12

$f - f_n$ 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 $n\to \infty$

Leaky Nun

how barbaric

to use $z$

ÍgjøgnumMeg

rofl

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

Leaky Nun

no!

it's $s = \sigma + it$

ÍgjøgnumMeg

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

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

Leaky Nun

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

this achieves nothing wait

Balarka Sen

03:18

@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.

ÍgjøgnumMeg

Nise one

Leaky Nun

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

Balarka Sen

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

ÍgjøgnumMeg

$\int \infty$

Leaky Nun

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

ÍgjøgnumMeg

03:20

The holomorphy?

Leaky Nun

yeah

ÍgjøgnumMeg

nise

Leaky Nun

the Morphy

ÍgjøgnumMeg

The hol got-damn morphy

I got a fun exercise to do tomorrow

Leaky Nun

cool

ÍgjøgnumMeg

03:22

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

Balarka Sen

@Leaky what was Morphy's defense again

just Ruy Lopez?

Leaky Nun

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

Balarka Sen

Ah

Leaky Nun

but I don't play the Ruy Lopez lol

Balarka Sen

most people just take the knight after that lol

messy pawn structure

Leaky Nun

03:24

> 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

Balarka Sen

yeah

I should read some book or something once in a while

Leaky Nun

same

Balarka Sen

I wanted to read Nimzowitsch's "My System"

but my blind chess playing friend said its trash lol

Leaky Nun

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

Balarka Sen

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

lol

Leaky Nun

03:28

lmao

talking about Short have you watched his famous game

Balarka Sen

the 7 games against "Bobby"?

Leaky Nun

the king march

Balarka Sen

yeah crazy

Leaky Nun

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

and when he found out, it was too late

lmao

Balarka Sen

that had to be some engine dude tho

Leaky Nun

03:30

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

Balarka Sen

Huh

Ted E

04:00

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

Any other recommendations?

Balarka Sen

@TedE Like Elder specifically, or something else?

They have a new EP which I also like

"Gold and Silver Sessions"

Very different, stylistically

Ted E

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

Balarka Sen

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

ÍgjøgnumMeg

HeRiTaGe

Balarka Sen

YES

ÍgjøgnumMeg

04:15

yaaas

Balarka Sen

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

ÍgjøgnumMeg

Not a clue, I love it

Heritage, Watershed, Blackwater Par

k

Balarka Sen

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

ÍgjøgnumMeg

Ah Ending credits is one of my favourite songs to play

rofl

Balarka Sen

It's great

I also like In My Time Of Need

ÍgjøgnumMeg

04:20

instagram.com/p/BTuWeOGhsH6

#Plug

Ted E

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

ÍgjøgnumMeg

wow 134 weeks ago

Balarka Sen

@TedE Ok so you actually do like heavier stuff

Cool @ÍgjøgnumMeg

Ted E

Some of it for sure yeah

Balarka Sen

I can reco you some atmospheric black metal

Ted E

04:22

Sounds good :D

ÍgjøgnumMeg

Eldamar's cover of Land of the Dead by Summoning

Balarka Sen

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

My top 3 albums

ÍgjøgnumMeg

Bergtatt is a masterpiece

Balarka Sen

Eldamar is great

ÍgjøgnumMeg

Do you know Winterfylleth?

Ted E

04:24

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

Balarka Sen

Nope, checking it out

ÍgjøgnumMeg

wait

Check out

Led Astray in the Forest Dark

by Winterfylleth

You will know this song hehe

Balarka Sen

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

ÍgjøgnumMeg

it is indeed lol

Balarka Sen

Oh ok it is a cover

ÍgjøgnumMeg

04:27

the song is an English translation tribute by Winterfylleth lol

Balarka Sen

Ahh

Balarka Sen

04:39

@ÍgjøgnumMeg This is pretty good!

ÍgjøgnumMeg

@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

Balarka Sen

Yeah we're followers of anti-Christ

ÍgjøgnumMeg

TrveKvlt

Balarka Sen

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

Sayan Chattopadhyay

04:41

Hahahaha

ÍgjøgnumMeg

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

Balarka Sen

lmao

black metal ist krieg bro

ÍgjøgnumMeg

Nargaroth all the way

Balarka Sen

its no joke

ÍgjøgnumMeg

I actually haven't listened to Nargaroth in ages

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

Balarka Sen

04:44

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

ÍgjøgnumMeg

A fine song

Oh btw, I saw Mayhem live the other day

Balarka Sen

Oh damn

ÍgjøgnumMeg

I gotta say

it was kinda lame

Gaahl's wyrd were much better

Balarka Sen

I haven't actually listened to too much Mayhem

I like the lore

ÍgjøgnumMeg

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

Balarka Sen

04:45

you know Necrobutcher said he wanted to kill Euronymous lol

ÍgjøgnumMeg

rofl

I was on my way to kill Euronymous too

Balarka Sen

Lmfao

ÍgjøgnumMeg

Did you watch Lords of Chaos?

Balarka Sen

Nope

Is it any good

ÍgjøgnumMeg

it's quite good

the actor who plays Varg is lame

Balarka Sen

04:46

I shall check it out

ÍgjøgnumMeg

but the film itself is great

Balarka Sen

nobody can play Varg I mean come on

ÍgjøgnumMeg

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

Balarka Sen

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

ÍgjøgnumMeg

shiet

se

c

Balarka Sen

04:50

heh its not trivial according to

oeis.org/A003172

ÍgjøgnumMeg

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

Ted E

How do you actually compute the class number in practice?

ÍgjøgnumMeg

5 is inert

Balarka Sen

minkowski bound or something

ÍgjøgnumMeg

oo 11 splits

fuck

Balarka Sen

04:53

damn

ÍgjøgnumMeg

@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

Balarka Sen

I tricked you actually

ÍgjøgnumMeg

although there are algorithms to solve those so

Balarka Sen

147 is 3 * 7^2

ÍgjøgnumMeg

oh fuk

Balarka Sen

04:56

sry

ÍgjøgnumMeg

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

Balarka Sen

lol rip

ÍgjøgnumMeg

actually that makes it easier

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

Balarka Sen

Ah

ÍgjøgnumMeg

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

I guess

which it does not

Balarka Sen

04:58

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

ÍgjøgnumMeg

I actually think you can still take that

Transcript for

$Mathematics$ Mathematics