Mathematics - 2014-06-18 (page 5 of 6)

@Pedro: You need to ask Kaj Hansen (a UGA undergrad) who's given some lectures (on YouTube, even) on Ramsey Theory. He answers a lot on main, and I'm confident he should handle this for you.

Pedro Tamaroff

@TedShifrin He doesn't seem to be around.

Ethan

@PedroTamaroff remember when you got me that book a while back

Pedro Tamaroff

@Ethan Which one?

Ethan

(removed)

Hippalectryon

Anyone knows where I can get Bourbaki, Algebra, Chap6, 2012 version online ?

Ethan

8:18 PM

"Mathematical Logic by Joseph R. Shoenfield"

do you think you could get it for me?

Chris's sis

@r9m This is subject to some new research. If I'm not wrong ...$$ \lim\limits_{n \to \infty} \sqrt[n]{ \operatorname{ LCM} \left( \binom{n }{0}, \binom{n}{1} ,\cdots,\binom{n}{n}\right)}=e$$

Ted Shifrin

@Pedro: I messaged him on Facebook.

Heya @Ethan.

Ethan

hi ted

r9m

@Chris'ssis $e$ ?! .. irk ,, gimmie some time to think .. :o

Ethan

(removed)

Alizter

8:21 PM

Hi @Ethan

Pedro Tamaroff

@Ethan Got it, gimme a second.

Ethan

@PedroTamaroff thanks

Chris's sis

@Ethan @r9m posted the question, not me. Well, it is subject to some research ... (from a certain point of view)

Ethan

lol

Hippalectryon

I meant chapter 8, 2012

Pedro Tamaroff

8:23 PM

@Ethan dropbox.com/s/nzs36zftmjb9qme/…

Ethan

@PedroTamaroff wow how do you get all this stuff lol

anyway thanks a lot

Studentmath

@Pedro maybe he means spanning subgraph indeed? That would make perfect sense

Ted Shifrin

@Yeah, @Studentmath, but he never says any such thing. I think this is likely to be a badly written treatise.

@Pedro: Kaj is thinking about it.

Studentmath

@Ted yeah, it should be made clear. But perhaps it is trivial for the author when he says 'subgraph'. Again, definitions..

Ted Shifrin

His examples of $2$-colorings certainly don't obey that requirement, @Studentmath.

Studentmath

8:27 PM

He never speaks of subgraphs there. Or does he?

Ted Shifrin

He uses the word. Funny, he belabors the pigeonhole principle, but then is sloppy immediately on the next page.

Alizter

@Chris'ssis I got it into this form but I am stuck $$\lim_{n\to\infty}\frac{\prod_{k=0}^{n+1}\binom{n+1}k}{\prod^n_{k=0}\binom nk}$$

Chris's sis

@Alizter There you have LCM.

Alizter

@Chris'ssis Yes, I know

r9m

@Alizter cool .. you got rid of the lcm .. how did you get to this form ?

Alizter

8:32 PM

$$\operatorname{lcm} \hat a=\frac{\prod_{k=0}^n a_k}{\gcd(\hat a)}$$

Pedro Tamaroff

Not, that's not true pal.

Alizter

where $\hat a$ is an ntuple

@PedroTamaroff Why not

Pedro Tamaroff

Try it for $n=3$.

Alizter

Ive been fed lies!

Ted Shifrin

@Alizter: We all take it as our mission to lie to you.

3

Chris's sis

8:35 PM

@TedShifrin lol

r9m

scratches head

Pedro Tamaroff

@TedShifrin Kaj is a lefty like me. +1

Ted Shifrin

Yes, he is, @Pedro. I know this from his having taken 3 classes from me (and not just from his video).

r9m

@Chris'ssis how did you get to the answer ? hints plz :) .. the fact that there is a da** lcm inside the $\sqrt[n]{}$ is taking my life :-(

Chris's sis

@r9m there is an (useful) formula for $$\operatorname{ LCM} \left( \binom{n }{0}, \binom{n}{1} ,\cdots,\binom{n}{n}\right)$$

Hippalectryon

8:40 PM

@Chris'ssis What's $LCM$ ?

Ted Shifrin

least common multiple, @Hippa

Hippalectryon

@TedShifrin ooh

Ted Shifrin

I actually don't know that in French :(

Hippalectryon

@TedShifrin $PPCM$

Ted Shifrin

et ça veut dire?

Hippalectryon

8:41 PM

plus petit commun multiple @TedShifrin

And yeah, the order of words IS reversed for the last two @TedShifrin

Ted Shifrin

hmm, in English word order, not French

:O

:D

Don't say that

I guess this is no place for French slang.

Hippalectryon

8:42 PM

Don't swear :c

r9m

@Chris'ssis should I try to get estimates for highest powers of primes in $\operatorname{lcm} \left( \binom{n }{0}, \binom{n}{1} ,\cdots,\binom{n}{n}\right)$ ?

or is there a cooler alternative formula ?

Chris's sis

@r9m $$\operatorname{ LCM} \left( \binom{n }{0}, \binom{n}{1} ,\cdots,\binom{n}{n}\right)=\frac{\operatorname{ LCM}(1,2,3,...,n+1) }{n+1}$$ and $\operatorname{ LCM}(1,2,3,...,n+1)$ behaves like $e^{n+1}$.

robjohn

@Chris'ssis Did you see the question about the GCD of all inner pairs on a row of Pascal's triangle? It is actually about $\gcd(n,\binom{n}{k})$, but the answer for the other question is given in the accepted answer.

r9m

@Chris'ssis ah .. how to prove that behaviour? :D

Alizter

I have no idea why the formula doesn't work and I want to learn algebra >:(

storms off

Chris's sis

8:47 PM

@r9m when $n\to\infty$ ...

@robjohn Where?

robjohn

@Alizter what is $\hat{a}$?

Chris's sis

@r9m prime number theorem?

r9m

@Chris'ssis oh nooooooo :-( .. is there no other way except ANTing ? :'(

Chris's sis

@robjohn nice (+1)

Ted Shifrin

@Alizter: What is the LCM of $4$, $6$, and $8$?

Welcome to @Kaj :) @Pedro, meet @Kaj; Kaj, meet Pedro. :)

Hippalectryon

8:53 PM

GTR isn't here today :c

robjohn

@Chris'ssis I only answered the question asked, but the general answer is very similar.

Ted Shifrin

Did you send him away, @Hippa?

Hippalectryon

@TedShifrin Shh don't tell anyone :P

Ted Shifrin

Méchant garçon. (One of my favorite lines from my favorite Jacques Tati movie.) @Hippa

Hippalectryon

@TedShifrin Old movies :C

Ted Shifrin

8:55 PM

As old as I am ... hush.

Pedro Tamaroff

I am leaving now. Be back in an hour or so.

Ted Shifrin

See ya, @Pedro. We are convinced that guy is wrong.

Chris's sis

3

Q: A formula for the least common multiple less than n?

Can anyone please tell me a exact formula for $\operatorname{lcm}(1,2,3,...n)$ and if its not possible then asymptotic expansion will do.

elementary-number-theory

I need a break since that integral put me down ... (temporarily I mean)

brb

Hippalectryon

@Chris'ssis Killed by a Titan seeking revenge ?

Chris's sis

@Hippalectryon :-))))))

robjohn

8:58 PM

@Chris'ssis an integral put you down? that doesn't sound like you

Chris's sis

@robjohn temporarily :-)

Ted Shifrin

Surely @Chris'ssis won't admit defeat to a silly little integral.

Chris's sis

@TedShifrin :D

Ted Shifrin

Someone asked me today whether $$\displaystyle\int \sqrt{x+\sqrt{x+\sqrt{x+1}}}\,dx$$ can be done in elementary terms. @Chris'ssis :D

Hippalectryon

mathoverflow.net/questions/171733/… @TedShifrin @Chris'ssis

Lol

Chris's sis

9:01 PM

@TedShifrin I asked something like that in the past ... let me see :-)))

Hippalectryon

So no it can't be done

Ted Shifrin

Right. I pointed him to Liouville's Theorem and gave him some references.

How did you find that so easily, @Hippa?

Hippalectryon

@TedShifrin I'm magic :D

Ted Shifrin

Zut alors.

Hippalectryon

The truth is

Ted Shifrin

9:03 PM

(You're not going to censor me on that one, are you?)

r9m

@Chris'ssis hugs at Chris's'sis' :-) .. life saver :') .. Chebyshev function is new to me .. :) thanks

Hippalectryon

I was just looking at the Math SO newsletter

And it was on it for this week

@TedShifrin Was this guy just testing you ?

Pedro Tamaroff

@KajHansen I didn't leave.

Ted Shifrin

I don't think so, @Hippa. But, not surprisingly, the person who answered that question is my brilliant old friend Robert Bryant (big star in differential geometry).

Hippalectryon

:D

Ted Shifrin

9:05 PM

I'll send him the link.

Kaj Hansen

Oh hello @Pedro. I didn't see that @Ted introduced us above. Nice to meet you.

Chris's sis

@r9m lollll :-) This chat is full of professionals, but sometimes you have the false impression is not like that. Many times I come here with fear and tremble ... :-))) (you never know when you're shot down :-)))) with a tough question)

Ted Shifrin

Maybe @Kaj doesn't ping ... too short?

Pedro Tamaroff

@KajHansen Well, it is only a partial meeting. =)

Ross has answered my question.

Ted Shifrin

Oh, you have an answer?

Kaj Hansen

9:07 PM

I do have ping actually. I didn't realize the sound was from the chat.

Ted Shifrin

Well, duh, so this is what we figured. He's tacitly assuming the subgraph is spanning. I wish the **** he'd say such things.

Hippalectryon

@TedShifrin math.stackexchange.com/questions/500589/… for the next level :D

Ted Shifrin

Very cool, @Hippa. Again, Robert :P

Pedro Tamaroff

@KajHansen As far is I know, $R(s,t)\leqslant R(s-1,t)+R(s,t-1)$.

But not with a $-1$.

Hippalectryon

@TedShifrin Yeah this problem is really cool ! i'm adding it to my pdf immediatly :D

Kaj Hansen

9:10 PM

@Pedro, I'm definitely familiar with the first result, but I could've sworn I saw the "-1" result in the literature somewhat recently. Searching now.

Here we go. Check out the 'note': en.wikipedia.org/wiki/Ramsey's_theorem#Proof_of_the_theorem

Ted Shifrin

@Hippa: Your .pdf? Now you need to learn some algebraic geometry and more :)

Pedro Tamaroff

@KajHansen Nice. My professor didn't prove the strengthened version.

Hippalectryon

@TedShifrin I've got a pdf where i keep track of some exercises i've done/met/...

Kaj Hansen

The unfortunate thing is that it doesn't hold in general. It holds only if $R(s-1, t)$ and $R(s, t-1)$ are even.

Ted Shifrin

Ah ... "Famous problems I have solved ... or wish I had ..." :D @Hippa

Pedro Tamaroff

9:13 PM

Oh, OK.

Hippalectryon

@TedShifrin I didn't say that xD

Pedro Tamaroff

@KajHansen I have a problem from my professor, that should use Ramsey Theory.

It is as follows.

Ted Shifrin

I kept folders in grad school of all the problems I solved while studying for qualifying exams. I still have those, but they're going in the recycle bin soon.

Hippalectryon

@TedShifrin send me a copy first :D

@TedShifrin WRONG LINK started may 16th 2014

Pedro Tamaroff

Given $m\geqslant 1$, prove there exists $n$ with the following property: for any binary $n\times n$ matrix (i.e. of $0$s and $1$s) there is a submatrix of size $m\times m$ such that all either all numbers in its diagonal are equal or all numbers under its diagonal are equal.

Hippalectryon

9:17 PM

@TedShifrin Here updated

dropbox.com/s/ay9yxp61kcdcjkw/mathsExos.pdf

Kaj Hansen

That definitely has a Ramsey-theory type flavor to it. We can reasonably assume that the smallest such $n$ exists when $m = 2$. Let me see if I can extract any insight.

Pedro Tamaroff

@KajHansen Well, I do have to leave now, but if you figure it out let me know and we can discuss it.

Kaj Hansen

Certainly. Nice meeting you @PedroTamaroff

Ted Shifrin

Bubye mr @Pedro

@Hippa: I'll have to think of some classic problems to give you ...

Hippalectryon

@TedShifrin :D

Ted Shifrin

9:23 PM

The problem is I don't know what you know and what you don't :D

Hippalectryon

@TedShifrin ask me :D

Ted Shifrin

Do you know multivariable calculus/analysis yet?

Hippalectryon

Nope

Like, dxdydz ? if so, not in maths

Ted Shifrin

Dommage ...

Hippalectryon

Just in physics

Ted Shifrin

9:25 PM

Un jour ...

Hippalectryon

:)

Alizter

What is in your opinion the best application for the isomorphism theorem for rings?

Ted Shifrin

Ah, my favorite is the proof that a prime $p=a^2+b^2$ for $a,b\in\Bbb Z$ if and only if $p\equiv 1\pmod 4$. @Alizter

Ironic that I said that since I don't like number theory :) But it's cool.

Hippalectryon

I've got one that's pretty cool but i don't have the proof yet

Ted Shifrin

You're doing ring theory, @Hippa?

Hippalectryon

9:28 PM

Prove that you can only partition a rectangle into rectangles of same area with an even number of triangles :D

Ted Shifrin

OH, that reminds me of one.

Hippalectryon

@TedShifrin No, it's not in the official PC's program - i like looking at the MP's however, and they study it

Alizter

I think the chinese remainder theorem is pretty cool :)

Ted Shifrin

Yes, it is, @Alizter, I agree.

Alizter

@TedShifrin I have been learning about ismorphisms and whatnot today. It took a while to get my head around quotient rings

Ted Shifrin

9:30 PM

@Hippa: This is sorta multivariable, but it's very cool. Suppose $f\colon\Bbb R^2\to\Bbb R$ is continuous and $\iint_R f dA = 0$ for every rectangle $R$ with area $1$. Prove that the average value of $f$ on the edges of every such rectangle $R$ is $0$. Then show $f=0$.

It takes some time, @Alizter, but very important concept.

lyme

sorry for interrupting @MartinSleziak thanks. can yuou tell me how can we optain $(0,1]$ and $[1,2) \in \tau_{left}\cup \tau_{right}$
I cant because my book defined these top. as $\tau_r:=\{R,\emptyset\}\cup \{(a,\infty)|a\in \Bbb R\}$ and $\tau_l:=\{R,\emptyset\}\cup \{(-\infty,b)|b\in \Bbb R\}$ so I think $\tau_{left}\cup \tau_{right}$ units should be like $(-\infty,b)\cup (a,\infty)$

Ted Shifrin

It's how you really understand arriving at $\Bbb Q(\sqrt2)$ or $\Bbb C \cong \Bbb R(i)$.

What??!! @lyme

Alizter

@TedShifrin I have noticed that it is really easy to whiz through aa and then get stuck.

Kaj Hansen

@Alizter, that's one of the things I love about the isomorphism theorem. It provides a way for looking at quotient rings without having to think about residue classes mod some ideal. That is, provided you are clever enough to think of a homomorphism with that ideal as its kernel.

Hippalectryon

@TedShifrin $\text{d}A=\text{d}x\text{d}y$ ?

Ted Shifrin

9:32 PM

Yes, @Hippa.

lyme

I see a notif. that martinsleziak replied me earlier. Cant I reply him like this?

blue

message sent from typewriter 6/18/2014 16:31 $\checkmark$

Hippalectryon

@TedShifrin Oh i've seen that much in multivars then :) i just haven't seen properly (ie in class) multivar derivatives and functions's theory

Ted Shifrin

@lyme, Martin is not often in here.

lyme

will he be informed if he is not in the chat?

Ted Shifrin

9:34 PM

Not sure, @lyme. It seems like you pasted a body of text, and it is not parsing correctly in here.

Alizter

@TedShifrin does it matter about $\Bbb R[x]$ or $\Bbb R(x)$ or is it squares for indeterminates and parenthesis for values?

Ted Shifrin

no, @Alizter, parentheses mean you make it into a field by taking quotients of polynomials.

r9m

@lyme ya sure ... link his last message to your next message (I mean reply to the last message he posted in this room) .. that will notify him :)

Alizter

@TedShifrin Wait... what?

@TedShifrin Never mind. I will get there eventually. I need to crack down on what I am learning now or else I will get no where. :)

Ted Shifrin

In fact, $\Bbb Q[\sqrt2] = \Bbb Q(\sqrt2)$ because, for example, $\dfrac1{1+\sqrt2} = \sqrt2-1$.

LOL ... ok @Alizter ... and no more snide comments about hearing me.

Alizter

9:37 PM

@TedShifrin Sorry professor.

Ted Shifrin

LOL, just teasing.

Hippalectryon

@TedShifrin i'm sure we could create some awful problem from the multinom's formula :D

Ted Shifrin

Not my style, @Hippa ... I don't do discrete math ... although I have to get good at it this fall.

Kaj Hansen

Ready for probability @TedShifrin?

Ted Shifrin

AGH, no. Wanna help, @Kaj? :D

Hippalectryon

9:39 PM

@TedShifrin Like a polynomial whose roots are the Vapnik-Chervonenkis dimension of a hypergraph of the set of coefficients of the rank n multinom formula :D

^ super idea

Balarka Sen

@TedShifrin Yuck that proof using abstract algebra.

Hippalectryon

i'm gonna be hated for generations if i do that :c

Kaj Hansen

Let's just say I'm not signed up for your class for a reason

Ted Shifrin

Yeah, @Kaj. You're fed up with me :D

That's a gorgeous proof, @Balarka. You hush.

Daniel Fischer

@TedShifrin How does it go (in broad strokes)?

Balarka Sen

9:41 PM

I'll hush ok. Just retorting about grumber theory, @TedShifrin

finite grumber theory is definitely not my thing.

(group + number = grumber)

Mike Miller

I haven't done math in a while. I'm afraid I've forgotten everything I once knew.

Alizter

@MikeMiller What is 1+2?

Ted Shifrin

@DanielF: $\Bbb Z[x]/\langle p,x^2+1\rangle \cong \Bbb Z_p[x]/\langle x^2+1\rangle \cong \Bbb Z(i)/\langle p\rangle$. :D

Mike Miller

I forgot the answer to that one a long time ago, @Alizter

Kaj Hansen

@Alizter, 0 mod 3?

Alizter

9:42 PM

@KajHansen obviously it's mod ?

Ted Shifrin

Heya @Mike. You getting old and stupid already?

Hippalectryon

4

@MikeMiller

Ted Shifrin

spanks @Hippa

Balarka Sen

@Hippalectryon HAHAHA

Kaj Hansen

Oh my goodness @Hippalectryon. That's golden!

Hippalectryon

9:43 PM

I knew this was gonna be useful :D

Ted Shifrin

@Hippa: You didn't get the English right!

glares @Kaj

Mike Miller

@KajHansen Ah, but with some simple inequalities and your result, one can show that $1+2 = 3 \in \mathbb Z$.

Balarka Sen

@Hippalectryon Hey, I was going to link to it.

Ted Shifrin

grumbles: I should never have allowed the videos

Hippalectryon

@TedShifrin Don't worry i still have 10+ to watch :D

Alizter

9:44 PM

@TedShifrin Just view it as good publicity!

Ted Shifrin

Publicity for what, @Alizter?

Balarka Sen

I can't stop laughing.

Hippalectryon

I won't watch all of them tho, only those i on subjects I don't know already lol

Alizter

@TedShifrin You know, for when you become king of the world?

Balarka Sen

That meme is so cool, @Hippa

Ted Shifrin

9:45 PM

Not happening, @Alizter.

Mike Miller

He's retiring from princedom soon, @Alizter, so no hope of kingship

Hippalectryon

memegenerator.net/Ted-Shifrin/caption

Ted Shifrin

Well, if you haven't had multiple integrals seriously, or differential forms or Stokes's Theorem, there's a lot to learn, @Hippa.

Je vais te tuer, @Hippa ... :D

fuis

fuis vite!

9:47 PM

Now I'll get banned for good.

Maybe that's a good thing.

Hippalectryon

:C

Nuu

tammam ulen uzatma ???

Mike Miller

@TedShifrin Is it obvious that 1-manifolds are smoothable?

I guess it is.

Hippalectryon

@TedShifrin It takes 3 hours to do Fubini's theorem ? i.gyazo.com/cc19eeb68660702f07f0959ed4b683ad.png

Mike Miller

Nevermind that question.

Ted Shifrin

9:54 PM

@Hippa: Proof was about 10 minutes. Lots of examples of the difference between iterated integrals and double integrals existing ... And then examples of switching orders of integration ... starting to visualize 3-D regions.

Hippalectryon

Ooh k

Ted Shifrin

Yeah, @Mike, any $1$-manifold is locally just an interval.

You're gonna be lots of trouble, @Hippa, I can tell. Time to put you on ignore soon. :D

Mike Miller

@TedShifrin I realized. I was trying to figure out why "well, just do it locally, and extend to its neighbors" doesn't work in general, and thinking about it for a bit made it obvious.

Ted Shifrin

So you're on vacation now, @Mike?

Hippalectryon

Seriously though, do you have like super quality chalk or boards ? I've never seen that i.gyazo.com/93d8bfbdf7c2b0575d5d3325022a1b48.png it seems like uber quality for a chalk board :3 @TedShifrin

Mike Miller

9:56 PM

@TedShifrin I was on vacation, now I need to start doing math again.

Ted Shifrin

These are new boards, @Hippa. We had our classrooms redone with sliding boards. They're actually not all that great. Wobble and erasure problems.

Mike Miller

Or else I really will forget all of it.

Ted Shifrin

LOL, @Mike. Grad school will put a stop to that in a hurry.

Mike Miller

@TedShifrin I'm reading a proof that all surfaces are smoothable. It's nice.

Hippalectryon