Mathematics - 2014-03-15 (page 1 of 3)

@Mike I've visited the Area 51 just now and on the left of Mathematics Educators appears the messages "1 day in beta" and the word "Visit!". Yesterday or the day before yesterday the word was "Commit" or "Commitment".

Pedro Tamaroff

12:15 AM

@Mike How do I mute that thing.

Mike

Mute your computer

Américo Tavares

... And clicking on "Visit!" one goes to the "Log in/Sign up" page, without further warning.

Pedro Tamaroff

@Mike But I am listening to music!

Fernando Martin

@Pedro: That solution is nice. You need to use that uniqueness theorem for ODEs though.

Pedro Tamaroff

@Mike I don't get it! I do one move and it all ends.

@FernandoMartin No, not really.

$f'^2+f^2=K$ and $f(0)=f'(0)=0$ gives $f=0$ identically.

No uniqueness necessary.

It's very elemental.

5space

12:20 AM

I like the original 2048.

Fernando Martin

You're right

Alexander Gruber

gahhhh

does anybody know where to find bourbaki's commutative algebra on the internet?

5space

@AlexanderGruber Which volume is it?

Alexander Gruber

@5space i'm not sure, this comment is why i'm asking

5space

I might have. Let me check.

@AlexanderGruber: That's like the only one I don't have pdf for.

Google has a preview: books.google.com/books/about/…

Maybe you can find it in there?

eXtremiity

12:35 AM

That game - is it the new craze?

Pedro Tamaroff

@5space Do you know how to play that game?

5space

Just the original.

I couldn't figure out the linked one :-/

Alexander Gruber

@5space aghghghgh

i need 12(e)

George V. Williams

@AlexanderGruber, I think I have that book.

Alexander Gruber

and that book skips a page and then resumes with 12(f) :(

Pedro Tamaroff

12:36 AM

@5space I think it's a troll thing.

George V. Williams

Yep. I'll post it as a comment.

eXtremiity

jennypeng.me/2048

Is this a troll ?

Alexander Gruber

@GeorgeV.Williams great, thanks

5space

@Pedro, I'm pretty sure.

eXtremiity

Can someone give me proper link to the game ?

Oh I see. What's your high scores?

Alexander Gruber

12:42 AM

@GeorgeV.Williams thanks. i think that's exactly what i need.

George V. Williams

@AlexanderGruber, always glad to help out, good luck.

Pedro Tamaroff

@eXtremiity Here

The one Mike linked to is a troll.

eXtremiity

Awesome. What's your high score?

Alexander Gruber

okay so first i need to show $A$ is flat over $A_i$

Pedro Tamaroff

@eXtremiity 3168. I am still getting a hold of the game.

eXtremiity

12:46 AM

Nice !

Jasper Loy

12:59 AM

It's a stupid game.

Karl Kronenfeld

Not sure how a game can be stupid/intelligent.

Pedro Tamaroff

@KarlKronenfeld Ask him how to change a light bulb.

Jasper Loy

It is 9 am, and I am waiting for the postman to come with my books.

Then I can go to sleep...

Pedro Tamaroff

My pen is acting up again.

Damn it.

Karl Kronenfeld

@PedroTamaroff The trick is to have 20 or so pens available at a time

Pedro Tamaroff

1:07 AM

@KarlKronenfeld Oh, but it is a fountain pen.

Alexander Gruber

@PedroTamaroff what kind?

Pedro Tamaroff

@AlexanderGruber A Parker Vector, Stainless Steel.

Alexander Gruber

@PedroTamaroff nice, i like parkers

@GeorgeV.Williams say, did you repeat a line in your transcription? there is an awkward change in the sentence that i am confused about.

George V. Williams

@AlexanderGruber, ugh, yes I did.

I was too concerned with the formatting t_t

Remove the second "there exists...".

Alexander Gruber

@GeorgeV.Williams is it correct now?

Karl Kronenfeld

1:18 AM

perhaps "there exists and index $\cancel{\alpha}$ $d$"

Pedro Tamaroff

@AlexanderGruber Would it be too much trouble for you to scan this=

Alexander Gruber

@PedroTamaroff i've read it. :)

Pedro Tamaroff

@AlexanderGruber It is good, yes?

Alexander Gruber

@PedroTamaroff yeah i liked it

it's got a lot of good stuff in there

George V. Williams

It should say "if E is a finitely generated submodule of $A_d$, there exists an index alpha and a finitely generated submodule $E_\alpha$ of $(A_\alpha)_d$ such that..."

Aside from that, @AlexanderGruber, it's good.

Joe Stavitsky

1:20 AM

how should I start on solving for x sqrt(x+3)+sqrt(x-1)=2?

Alexander Gruber

@GeorgeV.Williams how's that?

Karl Kronenfeld

the double subscript doesn't seem right

Alexander Gruber

@KarlKronenfeld i can't figure out what it means

maybe it's Bourbaki-talk for a localization?

Karl Kronenfeld

can't be

I'd guess it's supposed to be simply $A_\alpha$.

George V. Williams

@AlexanderGruber, earlier in the exercise, it says that "The right A-module A_d is coherent".

Alexander Gruber

1:24 AM

@GeorgeV.Williams hmm

Karl Kronenfeld

hm

George V. Williams

I'm just copying from the book, I haven't gotten around to actually learning from it yet, so I can be of limited assistance.

Alexander Gruber

@GeorgeV.Williams sure, i understand. i greatly appreciate what you're doing in any case

i'm just trying to figure out what that could mean, maybe it is some notation from the book

i wonder what would be a module over the direct limit?

George V. Williams

Here's the whole exercise: i.imgur.com/poZ99kF.png

Joe Stavitsky

btw, anyone have the math rendering link?

Alexander Gruber

1:27 AM

@GeorgeV.Williams i see, it's definited in exercise 11 apparently.

@JoeStavitsky it's on the sidebar.

@KarlKronenfeld I feel like it should be $E$ a finite submodule of $A$ and $E_\alpha$ a submodule of $A_\alpha$ with $E\cong E_\alpha\otimes A$.

George V. Williams

Here's exercise 11 then: i.imgur.com/XEQXGi0.png

Pedro Tamaroff

Not bad, 3412.

Karl Kronenfeld

20304

Pedro Tamaroff

@KarlKronenfeld Did you win? I am sometimes doing pretty fine, I think I have a decent strategy, and then I fuck up.

=D

Karl Kronenfeld

@PedroTamaroff Yeah, I won

Pedro Tamaroff

1:40 AM

@KarlKronenfeld Highest I got is 256, I think.

Karl Kronenfeld

@AlexanderGruber So exercise 11 is for the definition of coherent lol

gah

@PedroTamaroff Ah, what's your strategy?

George V. Williams

I got 256 by mashing buttons on my keyboard.

Probably just luck.

Karl Kronenfeld

@AlexanderGruber Here's an index of notation: books.google.com/…

found it

George V. Williams

@KarlKronenfeld, I have exercise 11 too: i.imgur.com/XEQXGi0.png

Karl Kronenfeld

@GeorgeV.Williams Exercise 11 defines coherent module and gives some properties. It unfortunately does not define $A_d$.

George V. Williams

1:44 AM

@KarlKronenfeld, ah, understood.

Alexander Gruber

@KarlKronenfeld i see that $A_d$ in the notation

Karl Kronenfeld

yeah, linking to the very beginning

Alexander Gruber

it says $\text{II}$ chapter 1 number 1

Karl Kronenfeld

ah, missed the chapter number lol

eXtremiity

Wooo !

6656 !

Alexander Gruber

1:47 AM

god damn it Bourbaki.

Karl Kronenfeld

For the life of me, I cannot find it

Ah, example 1

Yeah, it's just $A$

@AlexanderGruber So you're right here.

Pedro Tamaroff

@KarlKronenfeld Guess I can film one play. I got 4800 this time, and a 512.

Karl Kronenfeld

@PedroTamaroff Not gunna watch. :P

You can get surprisingly far without using the down arrow key

I think I used it twice when I beat the game.

Pedro Tamaroff

@KarlKronenfeld Hehe, OK.

eXtremiity

Oh, so you can BEAT the game?

Oh, a single 2048 beats it ?

Karl Kronenfeld

1:53 AM

Yeah, it stops after you get a 2048

eXtremiity

Just get all the big boys to the top left hand corner. Left -right left right to generate the small integers.

Then play accordingly to get the big boys together.

repeat.

I think =/

Its fun, no doubt.

Karl Kronenfeld

It doesn't really work that nicely

eXtremiity

That's why I didn't beat it :'( .

Karl Kronenfeld

It's often not worth while to put in the effort to get the big ones next to each other

Alexander Gruber

would it be so hard to just

write the definition in the notation section

Karl Kronenfeld

1:58 AM

lol

Alexander Gruber

@KarlKronenfeld wait i don't see it

Karl Kronenfeld

@AlexanderGruber Parenthetical remark at the end of example 1

Talking about how $A$ can canonically be endowed with the (one-sided) $A$-module structure.

Alexander Gruber

@KarlKronenfeld in what section?

Karl Kronenfeld

@AlexanderGruber I don't know their numbering and I closed out of it.

First section I think

Alexander Gruber

i don't see it

gahhh i'm going to commit seppuku

Karl Kronenfeld

2:02 AM

@AlexanderGruber See that big Z handwritten in the margin?

First example after that

Pedro Tamaroff

@KarlKronenfeld Not pressing down button is a good strategy. 7860.

Karl Kronenfeld

(This is the linear algebra chapter)

@PedroTamaroff Do note that there are times you should press down, it's just risky as hell

Pedro Tamaroff

@KarlKronenfeld It rhuins everything.

Karl Kronenfeld

@PedroTamaroff Yeah, you have to have a plan for combining all of the big ones and bringing them back up

Alexander Gruber

ugh whatever i give up

i'm just taking the d's away

Karl Kronenfeld

2:07 AM

@AlexanderGruber Shit there are page numbers.

192

Alexander Gruber

@KarlKronenfeld how did you get to that? it doesn't show it in the preview for me

Karl Kronenfeld

oh that explains it

no idea

@AlexanderGruber

Alexander Gruber

@KarlKronenfeld thanks

Karl Kronenfeld

I entitled the image bourbakisshittynotation.png

Probably uploaded as some other file name though

lol

Alexander Gruber

@KarlKronenfeld "to avoid confusion"

Karl Kronenfeld

2:11 AM

:D

Alexander Gruber

now for the hard part, actually proving that second part.

Pedro Tamaroff

@KarlKronenfeld 16484.

My set up was almost done.

Damn it.

I conclude this game should not be played with the down button. I mean, just pick three arrows and use those.

Winning strategy, pho sho.

eXtremiity

got my first 1024 !!!

yes, agreed

Use down ONLY when you have to initiate the big blow

i.e. combining the big boys

Alraxite

@PedroTamaroff I almost always use only 3 buttons!

Wait, we're talking about 2048, right?

eXtremiity

OMG IM SO PISSED OFF

:( .

Karl Kronenfeld

2:23 AM

@Alraxite Yeah

Alraxite

Cool.

I wonder anyone has ever got 4096.

eXtremiity

Yes, Karl has.

Karl Kronenfeld

No, the game ends once you get your first 2048

eXtremiity

Wait...

Alraxite

But you can keep going.

eXtremiity

2:25 AM

yeh xD oops.

Karl Kronenfeld

Shit, you can?

Well, damn

Alraxite

It gives you two options.

Well, it was implemented only recently.

Karl Kronenfeld

Maybe in an update

I only had the option to try again

Alraxite

Probably some hours back.

Karl Kronenfeld

yeah, I got 2048 yesterday I think

Alraxite

2:26 AM

I wonder how golden 4096 might be.

Ha.

Maybe it is red.

What was your score?

Karl Kronenfeld

20304

highest after 2048 was 32

so probably close to the minimum score

Alraxite

Ah, good. Nope I beat you with mine being only a 16.

Karl Kronenfeld

ah

Alraxite

And all rest were free 2s.

Karl Kronenfeld

heh, so not close to the minimum score eh. I had a couple 8s

Alraxite

2:29 AM

Well, on the plus side my board did look neat.

at the end.

so that's something.

A nice shining 2048 among white 2s.

Well, not really but I like to imagine it that way.

Karl Kronenfeld

heh

Alexander Gruber

2:42 AM

i really hate direct limit problems.

Pedro Tamaroff

@KarlKronenfeld I won. 20400.

I can keep going! Oh!

28976.

Pedro Tamaroff

3:16 AM

"We need a super-mathematics in which the operations are as unknown as the quantities they operate on, and a super-mathematician who does not know what he is doing when he performs these operations. Such a super-mathematics is the Theory of Groups." @AlexanderGruber

Nick

@Pedro: Wow, I never knew abstract algebra was that abstract.

Pedro Tamaroff

@Nick It's just a nice quotation, I don't think it conveys anything mathematically significant.

Fernando Martin

@Pedro: cms.dm.uba.ar/optativas/1ro_2014/model_sup_geom_alg

Pedro Tamaroff

Specially "... who does not know what he is doing when he performs these operations. "

Fernando Martin

Sounds cool

Nick

3:24 AM

Indeed it does

Pedro Tamaroff

@FernandoMartin Will you take it?

What courses did you sign up for?

Fernando Martin

It's just 2 weeks, it's a visiting prof.

Pedro Tamaroff

Oh.

Fernando Martin

I guess I'll take it if fits my schedule

Pedro Tamaroff

I have Linear Algebra's final in, but not Algebra II. =/

Fernando Martin

3:25 AM

You know the topics

That's all that matters

Real analysis, numerical calc, commutative algebra

Pedro Tamaroff

"Algebra conmutativa multigraduada, resoluciones libres, geometría de secciones." RUNS AWAY.

I don't know what a free resolution is. =P

Fernando Martin

I don't know what that stuff means either

Pedro Tamaroff

@FernandoMartin Oh. Well, I am learning pretty cool stuff about group theory. I dare to say a am "specializing" a little? =)

Fernando Martin

That's great

What are you studying?

Pedro Tamaroff

And I am not talking about modules.

@FernandoMartin At the moment, reading stuff about $p$-groups.

Fernando Martin

3:28 AM

I don't know much about GT, besides what I learned in Algebra II

Such as?

Pedro Tamaroff

@FernandoMartin Well, now learning stuff about the Frattini subgroup of a group, specifically of a $p$-group. It turns out that in such a case $G/\Phi(G)$ is an $\Bbb F_p$-vector space and every minimal generating set of $G$ (i.e. minimal wrt to inclusion) is also minimal w.r.t. to cardinality, which is equal to $\dim_{\Bbb F_p}G/\Phi(G)$.

Fernando Martin

What's a Frattini subgroup?

(or the?)

Pedro Tamaroff

Then comes stuff about the (in my opinion) correct approach to the FTFGAG.

@FernandoMartin The Frattini subgroup of a group is the intersection of all maximal subgroups.

It happens to be a characteristic subgroup, in particular normal.

Fernando Martin

Are you following Rotman's book?

Pedro Tamaroff

Yes.

Rotman kicks ass.

Except in some stuff.

Where he derps. =P

He has a wrong proof, in fact. At least me and two people agree so.

Mike

4:07 AM

Yo @Pedro

Pedro Tamaroff

@Mike Yiss?

Mike

Just sayin' hey

I took a huge-ass nap

Pedro Tamaroff

Good for you. =D

I should be sleeping.

Classes tomorrow morning.

Mike

Okie.

Pedro Tamaroff

@Mike I've answered some questions. You can look at those =D

@Mike Did you see the Von Mangoldt function one?

Mike

4:18 AM

You're better at number theory than I am.

Pedro Tamaroff

@Mike ORLY.

I don't even.

Mike

I'm not convinced the VM proof is better than the normal one...

Pedro Tamaroff

@Mike Damn right it is. =D

It is made to solve that kind of thins, really.

Mike

The other is really just straightforward counting, though.

Alexander Gruber

@PedroTamaroff yes! group theory power!

Pedro Tamaroff

4:30 AM

Haihai

Mike

hello alexander

Pedro Tamaroff

This is pretty easy, I guess: a p- group is cyclic iff its quotient by Frattini is.

Alexander Gruber

@PedroTamaroff $P/\Phi(P)$ is elementary abelian for $p$-groups

Pedro Tamaroff

@AlexanderGruber Yes.

It is the smallest subgroup with that property. =)

Thus must be $G'G^p$.

Mike

you're learning about this group from Rotman?

Pedro Tamaroff

4:36 AM

Yes.

@AlexanderGruber OK, maybe this one is more interesting: a group is nilpotent iff $G'\leqslant \Phi(G)$.

Alexander Gruber

so, here's one occurence of representations then: $P/\Phi(P)$ has an automorphism group isomorphic to $\operatorname{GL}_n(\mathbb{F}_p)$, where $n$ is the rank of $P/\Phi(P)$ as an elementary abelian group

Pedro Tamaroff

Yes. Burnside's Basis Theorem.

@AlexanderGruber So.

Suppose $G'\leqslant \Phi(G)$. Then $[G,M]\leqslant [G,G]\leqslant M$ for every maximal subgroup, so $M\lhd G$.

This means every maximal subgroup of $G$ is normal, so $G$ must be nilpotent.

Suppose now $G$ is nilpotent.

Then every maximal subgroup is normal (!), and has prime index.

This $G/H$ is abelian, and $G'\leqslant H$ for every maximal subgroup.

It follows $G'\leqslant \Phi(G)$.

Alexander Gruber

have you explored the connection between $\Phi(G)$ and $G^\prime$ and $F(G)$?

Pedro Tamaroff

@AlexanderGruber I think Rotman talks about the Fitting subgroup in exercises. I have to grow some bawls and look at it.

=)

Alexander Gruber

it's not scary. it's just the largest normal nilpotent subgroup of $G$.

Pedro Tamaroff

4:50 AM

@AlexanderGruber What connection between $\Phi(G)$ and $G'$ is there? I know that for any group $Z(G)\cap G'\leqslant \Phi(G)$ say.

I just showed $G'\leqslant \Phi(G)$ iff $G$ is nilpotent.

Alexander Gruber

i recall there was a bunch of exercises about those in isaacs, along with $F(G)$

i had at one point done them

Pedro Tamaroff

@AlexanderGruber Oh. I have a problem I couldn't finish. =I

I want to show that a group is (1) nilpotent iff (2) $H<G$ proper implies $H<N(H)$ (the normalizer condition) iff (3) $M<G$ maximal implies normal.

Now, I have it almost done.

For example, (1) iff (2) by Sylow stuff.

And (2) implies (3) is immediate.

So I need to either show (3) implies (2) or (3) implies (1).

Alexander Gruber

3 => 1 by contrapositive might be good

Pedro Tamaroff

How?

Alexander Gruber

if it's not nilpotent, at least one sylow subgroup isn't normal, yes?

Pedro Tamaroff

4:59 AM

Ah, indeed.

@AlexanderGruber I claim $P=N(P)$.

I am trying to find a subgroup of $G$ that is self normalizing, @AlexanderGruber

Pedro Tamaroff

5:26 AM

Well, if anyone comes with a solution, let me know.

Mike

5:49 AM

Hahahaha

@PedroTamaroff there's unintentional humor there.

Mike

6:27 AM

@KarlKronefeld tell me when you're around

skullpatrol

7:53 AM

0.999...hour later...

Utkarsh

8:04 AM

@skullpatrol sorry

Sawarnik

@Utkarsh What for?

Utkarsh

he knows ;p

@Sawarnik so, how is your python running?

Sawarnik

Hmm...you have learnt things.

Parth Kohli

...it has your phone number. delete.

Sawarnik

@ParthKohli You are very quick at spotting those. Thank God you are not the NSA :)

Parth Kohli

8:08 AM

@Sawarnik I was just about to say that... haha

Sawarnik

Lets move to the other room.

Karl Kronenfeld

9:26 AM

@Mike You misspelled my name, so I didn't get pinged...

@Mike Anyway, 'sup

Chris's sis

11:07 AM

Greetings

It's similar to the result I posted yesterday.

Pedro Tamaroff

11:22 AM

@Mike OH SNAP!

Well if the solution is that good, certainly share it.

Chris's sis

11:40 AM

@Sawarnik I had a comment about that in the past -> math.stackexchange.com/questions/178883/…

usukidoll

0

Q: Proving isomorphisms from posets.

An isomorphism from a poset $(S_1,R_1)$ to a poset $(S_2,R_2)$ is a bijection $f: S_1 \rightarrow S_2$ such that, for all $x,y \in S_1$ $(x,y) \in R_1 \leftrightarrow (f(x), f(y)) \in R_2$ When such an isomorphism exists, we say that $(S_1,R_1)$ is isomorphic to $(S_2,R_2)$. Questions: S...

@Utkarsh np

@skullpatrol Hi

@IceGirl Hi

@skullpatrol Please come to ELL

skullpatrol

11:55 AM

@IceGirl OK

Transcript for

$Mathematics$ Mathematics