Mathematics - 2013-07-31 (page 1 of 2)

Peter Tamaroff

12:06 AM

@anon Hello?

anon

yes

you were going to ask a question?

Peter Tamaroff

@anon I thought you were going to give me your opinion.

anon

about taking LA instead of A-II?

Peter Tamaroff

@anon Not really. About taking LA alone, and not both LA and A-II.

anon

sure, go ahead.

Peter Tamaroff

12:09 AM

@anon OK?

=)

Peter Tamaroff

12:27 AM

Let's be realistic/mean: why would one be studying Elliptic curves without knowing how to depress a cubic?

user1

(Continuing the realistic/mean pattern) I am not sure I would recall immediately the transformation, but in the time it takes to write that question I would have figured it out.

mick

How to simplify the ratio of 2 integrals from 0 to oo ?

Peter Tamaroff

@user1 Heh, true.

mick

int 0..oo f(x) dx / int 0 .. oo g(x) dx = ???

@PeterTamaroff

Peter Tamaroff

@mick "INSUFFICIENT DATA FOR MEANINGFUL ANSWER."

mick

12:41 AM

assuming the integrals cannot be done

@PeterTamaroff looking for a strategy , no specific in mind

Peter Tamaroff

@mick "THERE IS AS YET INSUFFICIENT DATA FOR A MEANINGFUL ANSWER."

mick

I mean can it be rewritten as a single integral ?

Peter Tamaroff

@mick Not knowing what $f$ or $g$ are, who knows!

mick

@PeterTamaroff thus , not in general ?

Peter Tamaroff

Of course, suppose the integrals exists, and their quotient is $\ell$. Then $$\ell =\int_0^1 \ell dx$$

Or $$\ell=\left(\int_0^\infty e^{-\ell x}dx\right)^{-1}$$

But that is boring.

mick

12:46 AM

yes i mean an intresting way :)

hmm i can prove it myself nvmd

OMG!!

this is amazing !!

Charlie

@amWhy im glad tô know that, :)

amWhy

@Charlie I'm a happy camper, when all is said and done :^)

Charlie

@amWhy :D

@mick hi

:( no one says hi

anon

@Charlie hi

Charlie

ONLY amy

@anon Pedro is here too, you can say hi to him

mick

12:57 AM

@Charlie hi

Peter Tamaroff

@Charlie We've already saluted accordingly.

anon

@Charlie I am unable to conclusively decipher the mood of this comment.

mick

my mentor is amazing !!

OMG

Peter Tamaroff

@anon Bitches be crazy.

anon

hey now

no need for that kind of talk

mick

12:59 AM

@anon she missed me

Charlie

@anon you and pedro are so smart...

anon

:/

Peter Tamaroff

@Charlie I am only passionately curious.

mick

Look what my mentor wrote

its amazing

Charlie

@mick who's your mentor? Obi-wan?

mick

1:01 AM

@Charlie worse than chuck norris ! :)

Peter Tamaroff

@anon ¿Por qué esa mueca?

mick

check this out : @Charlie

http://math.eretrandre.org/tetrationforum/showthread.php?tid=799

=)

@mick hmm...

1:03 AM

@Charlie its tommy1729 see link.

Charlie

@PeterTamaroff hahahaha

mick

Did you see the link ? @Charlie ??

Charlie

Yes

mick

He writes equations like a boss , at least imho

Charlie

Hmmm

mick

1:07 AM

although tex would be nicer

Isnt that the most amazing equation ever ?? @Charlie

Charlie

@mick no, sorry

mick

@Charlie you seem not impressed.

Charlie

@mick yeah...

mick

@Charlie weird

i find it amazing and beautifull

Charlie

Beauty is in the eyes of the beholder

mick

1:12 AM

@PeterTamaroff whats your opinion about all this ?

A few weeks ago , my mentor was doing integrals out of his head ... while playing chess and reading russian !

Charlie

Wow...

mick

ill never be that good :/

Peter Tamaroff

@mick I own a chimpanzee that can count up to 10.

mick

@PeterTamaroff dont be racist to james , your black butler :p

robjohn

@AntonioVargas I don't know of a definition that I could call universal. I would say it depends on the context

Peter Tamaroff

1:16 AM

@mick That is offensive.

mick

@PeterTamaroff srr

Peter Tamaroff

I will proceed to ignore you.

mick

@PeterTamaroff said srr

Charlie

I can't talk math like @anon and @peter but I believe that integras must at least have a meaning, otherwise in my opnion, it wel be only an integral.it should have context

robjohn

@Charlie and I've seen some pretty ugly eyes :-D

Peter Tamaroff

1:18 AM

@robjohn BADUM TSS!

Charlie

@robjohn XD

robjohn

Gotta get to the park. BBL

mick

@robjohn not impressed either ?

you guys are hard to impress.

robjohn

@mick not sure what you mean. I was making a joke about a time-worn expression.

Peter Tamaroff

@mick I read nonsense, mick. Come on, be realist.

mick

1:20 AM

its about this (imho amazing !)

http://math.eretrandre.org/tetrationforum/showthread.php?tid=799

@PeterTamaroff nonsense ??

Peter Tamaroff

@mick Yes. $\log\log\cdots \log x$ is undefined for every $x$.

mick

@PeterTamaroff but iterations of arcsinh(x/2) are not.

Peter Tamaroff

@mick I read the nonsense first. Was taken aback.

mick

@PeterTamaroff read on

Peter Tamaroff

Sure, $\sinh^{-1}\cdots\sinh^{-1}x$ converges to zero for any $x$.

mick

1:25 AM

@robjohn its about this (imho amazing !)

math.eretrandre.org/…

Peter Tamaroff

@mick What is amazing about that?

Charlie

Ow..

mick

@PeterTamaroff a power law related to logs ??

@Charlie what ?

Peter Tamaroff

@mick Power law?

Charlie

@mick it's is amazing/ not it's not..../ it is/ no/yes/no

mick

1:29 AM

@PeterTamaroff signomial

@Charlie are you ok ?

Charlie

@mick why you ask?

mick

@Charlie no/yes/no ????

Charlie

@mick your chat with pedro

mick

Well i find it amazing and cool

Charlie

Ok, good. That's it.

mick

1:32 AM

who's Pedro ? I talk to Peter "griffin" Tamaroff :)

Charlie

Pedro nicoláaaaaas

mick

nobody likes my master :/

everybody loves me :)

Charlie

Everybody loves Raymond

mick

and chuck norris.

Charlie

I must go

Bye, you all

mick

1:42 AM

bye. im off too

bye@PeterTamaroff @robjohn

Charlie

Bye

Ian Mateus

2:18 AM

Is it my browser or many pictures are dead by now?

Peter Tamaroff

@IanMateus Same here.

Ian Mateus

It seems imgur is the problem

@PeterTamaroff randomly click in a broken image and in a non-broken one. They probably are from different places

Here

It is normal now. What was that?

Peter Tamaroff

@IanMateus Elemental, Whutsawn

Peter Tamaroff

3:00 AM

@EricTressler Yello.

What brings you to $\Large{\mathscr{The}\;\;\mathscr{Honorable}\;\;\mathscr{Chat} }$?

Eric Tressler

5:34 AM

Hi

Martin Sleziak

6:01 AM

Why was this post deleted? Comments layout break if Math expression is too long

Other post was closed as a duplicate of this one. So it does not seem as a good idea to delete it...

I cannot vote to undelete: A moderator has deleted this post and it cannot be undeleted.

Any mods here?

Eric Tressler

i tried to look to help explain, but i can't even see it, since it's deleted

but from the title. it sounds like a LaTeX problem, and not a math problem

so it would be off-topic for math.stackexchange

Martin Sleziak

@EricTressler That's why it is posted on meta.

(Where it belongs.)

Eric Tressler

sorry, i'm only trying to be helpful. I only get a dead link.

Martin Sleziak

Deleted links are visible for mods and 10k+ users, but it seems that only mods are able to undelete this one.

Eric Tressler

do you think it shouldn't have been deleted?

Martin Sleziak

6:11 AM

I've posted the message here (maybe the two mods in this room will notice it) and I've also left a comment on a post of another moderator.

Eric Tressler

you still didn't answer me

Martin Sleziak

Hopefully some of them will notice and explain or correct this.

I am too slow typer obviously.

I think it should not have been deleted. If not for other reason, then because of the fact that another post has been closed as a duplicate of this one.

Eric Tressler

well, what was your question

Martin Sleziak

So we now have a post marked as duplicate of another post, but that post is deleted.

Tobias Kildetoft

@EricTressler you are coming across as really slow right now

Eric Tressler

6:13 AM

i see. I can't help with that, but maybe I can help with your question

Tobias Kildetoft

he said all this further up

Martin Sleziak

My question was why the post was deleted. And it was addressed mainly to the mods.

The (deleted) question I linked to was posted some time ago by another users and it has been already answered.

Eric Tressler

@TobiasKildetoft I don't see any of that, and so I don't care

Martin Sleziak

It was about the problems with comments which contain too long MathJax expressions and they cannot be deleted.

Several questions tagged comment+tex are about the same problem.

In this one Willie Wong posted a workaround using scaling math expressions in MathJax settings.

Eric Tressler

Oh; I see. I don't think even the moderators can do much about that. It may be an inherent flaw in the stackexchange framework

can you avoid it using eqnarrays or the like?

Martin Sleziak

6:18 AM

It usually happens when a user makes a typo, I'd say. For example unpaired dollar.

Example of such comment is here.

Eric Tressler

Well, I don't think the moderators have much to do with this. It's just a software limitation that you'll need to deal with unless enough people get annoyed by it that the designers change the specs

which is to say, effectively never

Martin Sleziak

Well, but definitely the moderators are the ones to contact if you think that a post have been deleted and it should not have been.

Eric Tressler

yes

Martin Sleziak

As I said, this was extensively discussed at meta (you can find several posts about it).

Eric Tressler

but you should have some good reasons why it should remain, and they'd probably give you the benefit of the doubt

Martin Sleziak

6:23 AM

Since there are two moderators logged in this room, I thought posting a message here would be a good way to making them aware of the problem.

Eric Tressler

well, they aren't answering, but I am. what's the problem?

Tobias Kildetoft

This is getting rather comical

2

Martin Sleziak

They don't have to answer right away. They're mods, not a call center.

Eric Tressler

You claim something's wrong, but you can't elucidate me?

Tobias Kildetoft

@MartinSleziak do you think he is trolling?

Martin Sleziak

6:26 AM

Since they hang around in this chatroom, they probably have a look at messages posted while they were AFK, when they get back.

Eric Tressler

OK. What is your problem?

Martin Sleziak

@TobiasKildetoft I have doubts that someone having close to 1k reputation would be trolling?

@EricTressler The only thing I wanted to make was to make mods aware of the problem. (At least I think it is a problem.) I think I have done this. (And if I see one of the mods active in chat when I come back later, I'll ping them to ask whether they have seen what I've posted here.)

So I don't think anything can be done until one of the mods responds.

Maybe we could leave it at this. I really should go and take a shower and then take a walk to work.

Eric Tressler

@MartinSleziak: Do you think that's a mathematical question?

Martin Sleziak

Thanks for your interest, anyway.

@EricTressler No, it definitely is not.

But this room has commonly been used to contact mods or other users, which are in chat for all kinds reason.

Tobias Kildetoft

@MartinSleziak you still think he is not trolling?

Martin Sleziak

6:32 AM

I don't think that his message was a reason to flag, I can live with someone calling me a name. (To whoever flagged that....)

Tobias Kildetoft

@MartinSleziak that was me. I felt his general behavior was trollish and that last message was the last straw

2

Martin Sleziak

@EricTressler It was definitely not my intention to waste you're time. As I said, only mods can do something about that problem and this conversation is going nowhere. As I don't want to waste might time either, I'll leave the chatroom now. (I might come back later today.)

I'd be grateful if any of the mods @robjohn @MarianoSuárez-Alvarez could unfreeze the Jury Duty chatroom.

The problem I've mentioned above would be more suitable there and perhaps there would not be confusion why I am posting this to the main chatroom.

@TobiasKildetoft I think we have seen much worse in this chatroom.

Anyway, I'm leaving, see you later (maybe).

N3buchadnezzar

8:04 AM

yo

skullpatrol

9:24 AM

yo

9000 hours later...

N3buchadnezzar

9:47 AM

Landen's transformation

Landen's transformation is a mapping of the parameters of an elliptic integral, which leaves the value of the integral unchanged. It was originally due to John Landen, although independently rediscovered by Carl Friedrich Gauss. In Gauss's formulation, :I = \int _0^{\frac{\pi}{2}}\frac{1}{\sqrt{a^2 \cos^2(\theta) + b^2 \sin^2(\theta)}} \, d \theta is unchanged if \scriptstyle{a} and \scriptstyle{b} are replaced by their arithmetic and geometric means respectively, that is :a_1 = \frac{a + b}{2},\qquad b_1 = \sqrt{a b}.\, Proof The transformation, may be achieved purely by integration...

robjohn

10:12 AM

@MartinSleziak a developer deleted it, and I am discussing it now.

The question itself has been answered and worked around in this question and this question.

Martin Sleziak

@robjohn Thanks!

robjohn

@MartinSleziak no problem.

Martin Sleziak

@robjohn I think these questions are different. The questions you linked are about mathjax broken due to added whitespace and the question I have linked to was about long formulas, which would make difficult to delete the comment, because delete button is hidden somewhere behind other stuff.

robjohn

@MartinSleziak I guess I never saw that question and it was never associated with the qustions I answered.

Martin Sleziak

The workaround suggested by Willie in the deleted question was to scale fonts in MathJax settings.

robjohn

10:16 AM

@MartinSleziak Ah...

@MartinSleziak Yes, I think I answered a question like that recently, too :-)

Martin Sleziak

Another workaround I know about would be to reload a page and be quick enough to delete comment before MathJax gets rendered. Slow internet connection is preferable for this one.

@robjohn Yes, such questions have been certainly asked and answered several times. But I did not like that this one was deleted because: 1) another question is closed as a duplicate of it; 2) I did not see Willie's workaround elsewhere; 3) there is (at least one) answer to other question pointing to this one.

Anyway, thanks for looking into the matter. I'd be glad to know how it is resolved. So feel free to ping me even if I am not in chatroom at the moment.

robjohn

@MartinSleziak There, I found it :-)

@MartinSleziak Ah, which question points to the deleted one? That will prove its importance.

Martin Sleziak

A moment, I'll find it.

robjohn

@MartinSleziak here

Martin Sleziak

Yes, this is the one closed as duplicate.

And Willie's answer here also has a link to that question. (I've added a comment there mentioning it.)

robjohn

10:30 AM

@MartinSleziak I was just about to mention that one

@MartinSleziak and here

Martin Sleziak

@robjohn Yes, I've added the comment today. (I did not notice that it was deleted at first, but the OP of that question is 10k+ anyway.) That's why I've noticed the problem and try to contact the mods.

robjohn

So I can cite 1, 2, and 3.

Martin Sleziak

Yes.

Although my comment was added only after the question was deleted, so it does not probably have that much weight.

robjohn

@MartinSleziak I just won't mention that ;-)

Martin Sleziak

:-)

robjohn

10:35 AM

@MartinSleziak The dev hasn't been active for over half an hour, so it may take a while.

Martin Sleziak

@robjohn Well, it's definitely something that has to be resolved immediately.

robjohn

:-p

Martin Sleziak

I am glad to know that at least one of our mods is aware of it. And I guess it will be resolved one way or another eventually.

robjohn

@MartinSleziak yeah. I will let you know, but keep an eye on that question.

in case I forget (me, forget? never!)

Martin Sleziak

I've left a comment to Willie Wong. (Basically with the information that one of the mods already knows about it.)

robjohn

10:40 AM

@MartinSleziak done (unfrozen).

Martin Sleziak

Oh, it's already undeleted.

Ok, I can delete my obsolete comments.

robjohn

@MartinSleziak no...

I linked to the question regrding the Jury Room

Martin Sleziak

Oh, I see...

Thanks for that!

BenjaLim

hell this sheaf theory shit ton of indices all over the place

robjohn

@BenjaLim I knew there was a reason that I avoided it.

Willie Wong

12:57 PM

@MartinSleziak JFYI, robjohn noticed because I pinged Oded in the mod chat room after you left your first comment. When Oded dropped by robjohn was the only one home.

Martin Sleziak

ok

Nick

Look what I found: A proof for Legendre's conjecture vixra.org/pdf/1303.0048v1.pdf

Martin Sleziak

I guess I can delete my comments I left at your answer to ping you, Willie.

Willie Wong

@MartinSleziak yes indeed. * wink *

Tobias Kildetoft

@Nick and on vixra no less. Then it must be correct

Martin Sleziak

12:59 PM

@WillieWong And I see that the question is undeleted.

Was it deleted by mistake somehow?

Nick

@TobiasKildetoft: was that you you being sarcastic or are you genuinely saying that the Legendre's conjecture is true?

Tobias Kildetoft

@Nick that was me being sarcastic

Willie Wong

Sort of. Oded was cleaning up the bug tag. Since there was no "official action" and MathJax was updated like 3 times since that question was posted, and since there have been no recent action on that question (no new comments or answers), he assumed that the bug was fixed on the MathJax end.

Martin Sleziak

ok

I think your answer contains useful information, so I am glad it is visible again.

BTW congrats on getting tex badge on meta.

Nick

@TobiasKildetoft: awww. too bad.

Tobias Kildetoft

1:04 PM

@Nick I wonder if anything worthwhile has ever been posted on vixra

Willie Wong

I've sent him some possible improvements that the site can use in regard to that bug. (Which is something about how MathJax interacts with the site, so sort of tricky). But if it is something that is out of their control then it will either be marked -bydesign or -wontfix, probably soon.

@MartinSleziak thx

Next target: a gold topic badge on Meta... :-)

Nick

@TobiasKildetoft: Why is it that when you can clearly observe something to be true, there can still exist no proof for it.

Martin Sleziak

I thought that deletions and undeletions are list on the timeline, it seems I have been mistaken.

Tobias Kildetoft

@Nick what can you observe to be true that there is no proof for?

Nick

@TobiasKildetoft: Legendre's conjecture ofcourse. I've seen primes in between squares upto $943^2$ and $944^2$

Martin Sleziak

1:11 PM

Gold tag badge on meta seems a rather ambitious goal (perhaps with the exception of the compulsory tags).

Anyway, thanks for letting me know that the issue with that question has been resolved @WillieWong

Willie Wong

@MartinSleziak It is because you are not a mod. The mod display for the timeline does include deletion information.

@MartinSleziak I am pretty close on (discussion) :-p

Tobias Kildetoft

@Nick but those are not very large numbers of course.

Martin Sleziak

@WillieWong Thank god I am not a mod....

Willie Wong

Alright, back to work.

Martin Sleziak

But good to now that it is displayed somewhere.

Nick

1:15 PM

@TobiasKildetoft: I guess but if there exists a prime between every n and 2n. There must surely be one for every n an n^2

Tobias Kildetoft

@Nick you mean between $n^2$ and $(n+1)^2$ and I agree, it seems like it should be easier than Bertrand

Nick

@tobiasKildetoft: yes sorry, and yet it still remains a conjecture even after Bertrand's has been proved.

Frank Science

1:54 PM

@TobiasKildetoft It's quite harder than Bertrand.

Tobias Kildetoft

@FrankScience yes, I realize that, or it would have been proven

Nick

Well, it took billions of years for humanity to give a proof for 1+1 = 2. I guess this is just one of those things

Frank Science

I have trouble in proving the easy half of the Quillen-Suslin theorem. It states that if $V$ is a free module over $\mathbb C[x_1,\dotsb,x_k]$ whose rank is $r$ and $A$ is a $m\times n$ presentation matrix of $V$, then $A(c)$ has rank $m-r$ at every point $c=(c_1,\dotsb,c_k)$.

BenjaLim

2:42 PM

@robjohn I am listening to this. youtube.com/watch?v=gB8T0dplrU0

robjohn

3:21 PM

@BenjaLim Nice. I don't think I've listened to that before.

Eric Tressler

3:49 PM

The "1+1 = 2" thing is a little strange; few people outside are as careful as Russell and Whitehead

er, outside of logic, I meant

David Wheeler

4:06 PM

@BenjaLim stuff like what you linked to always puts me in mind of Pachebel's canon... but with a beat.

N3buchadnezzar

yo

David Wheeler

what do you mean, @EricTressler?

Eric Tressler

I was just commenting about @FrankScience's remark

oh, no, it was @Nick

anyway, relatively few mathematicians concern themselves with foundational problems like that

David Wheeler

i've always felt 1+1 = 2 is more "definitional" really, more of a name-game...2+2 = 4 has much more "meat" to it.

Eric Tressler

@David youtube.com/watch?v=WvDedo8r1Fk

N3buchadnezzar

4:13 PM

@robjohn You here ? =)

Frank Science

Well, I'll post a question.

N3buchadnezzar

I have a differential equation I have some problems solving

$$ \nabla\left( I_n(a,b) \right) = (1-n) I_n(a,b)$$ with $I_1 = 2\pi/(ab)^{1/2}$. Bah

N3buchadnezzar

4:28 PM

Hmmm, soo close to solving it

It has something to do with the binomial formula..

Seems like

$$I_n = \frac{ \frac{\partial^{n}}{\partial a^{n}} + \binom{n}{n+1} \frac{\partial^{n} }{\partial b \partial^{n-1} a} + \cdots}{(n-1)(n-2) \cdots (n-n+1)} $$

Omnitic

4:44 PM

Anyone know a proof if ab=cd then a+b+c+d is not prime?

for positive integers a,b,c and d

Chris's sis

Greetings

N3buchadnezzar

@Chris'ssis Yo

Chris's sis

@N3buchadnezzar hey :-) How is it going?

N3buchadnezzar

@Chris'ssis Good, was stumpled by a problem.. See above, you?

Chris's sis

@N3buchadnezzar yeah. Unfortunately I'm in the middle work for $$\sum_{k=0}^{\infty} \frac{2^{2k} B_{2k}}{(2k+2)!}$$

N3buchadnezzar

4:54 PM

@Chris'ssis Looks like arctan or something ?

Chris's sis

@N3buchadnezzar which problem do you refer to?

N3buchadnezzar

@Chris'ssis Oh, your sum reminds me of the taylor expansion of arctan

Chris's sis

I see.

robjohn

5:19 PM

@N3buchadnezzar I am. Sorry, I was answering a question.

N3buchadnezzar

)

robjohn

5:30 PM

@Chris'ssis Are you using $$\frac{t}{e^t-1}=\sum_{k=0}^\infty\frac{B_kt^k}{k!}$$

Chris's sis

@robjohn I was thinking to use $$\int x \coth(x) \ dx = \sum_{k=0}^{\infty} \frac{2^{2k} B_{2k}}{(2k+1)(2k)!} x^{2k+1}$$

I'll continue it after doing some jogging. :-)

robjohn

@Chris'ssis That looks pretty close to what I gave after one integration. You'll need two.

Chris's sis

Yes.

@robjohn today I met a shocking beautiful question. I'll post it here before leaving.

N3buchadnezzar

5:51 PM

Hi

Chris's sis

$$\lim_{n\to\infty}n \int_{0}^{\pi/2} \cos^n(x) \sin(n x) \ dx$$

N3buchadnezzar

The answer is something like the avreage value of $cos(x)$ over $(0,\pi/2)$ me thinks.

@robjohn Can one write $$ \left( \frac{\partial }{\partial a} + \frac{\partial}{\partial b} \right)^2 f(a,b) $$ ? The notation seems somewhat funky :p

robjohn

@N3buchadnezzar Certainly; you have done it. QED

N3buchadnezzar

Well, what would you think it means?

robjohn

@N3buchadnezzar $\left(\frac{\partial^2}{\partial a^2}+2\frac{\partial^2}{\partial a\partial b}+\frac{\partial^2}{\partial b^2}\right)f(a,b)$

N3buchadnezzar

5:58 PM

=d

Working on an answer :p

Peter Tamaroff

@Chris'ssis Let me look at it for a second.

Chris's sis

@PeterTamaroff OK. I'll answer you back later. I go out right now.

Peter Tamaroff

@Chris'ssis I have an idea.

N3buchadnezzar

6:28 PM

@PeterTamaroff Hey

Peter Tamaroff

@N3buchadnezzar What be oozin?

N3buchadnezzar

@PeterTamaroff Could you check over an answer of mine? I am a bit unsure if I got the indexes right.. Whether it should be $n$ or $n+1$ =)

Peter Tamaroff

OK.

@N3buchadnezzar What is it you are not sure about?

N3buchadnezzar

The gamma part, It might be n! instead.

Alizter

why do you use mathjax in chat when it doesn't work?

N3buchadnezzar

6:35 PM

The part to the differential equation is 100% correct, the part after "it can be shown by induction"...

@Alizter Cause real men can read plain latex.

@Alizter Psst mathjax works in chat, look to the right

Peter Tamaroff

6:46 PM

@Chris'ssis Since for any $\delta >0$ $n\cos^n x$ converges to zero uniformly on $[\delta,\pi/2]$ it suffices we look at the integral over $[0,\delta]$ for some small $\delta>0$. We want to use that $$\log\cos x\sim -\frac{x^2}2$$ near to origin, so we choose a small $\delta>0$ so this approximation is indeed accurate, say $$ - \frac{{{x^2}}}{2} - \varepsilon < \log \cos x < - \frac{{{x^2}}}{2} + \varepsilon $$ for $\epsilon$ the desired accuracy.

Then we look at $$n\int\limits_0^\delta {{e^{ - \frac{{n{x^2}}}{2}}}\sin nxdx} $$

This converges to $1$.

It remains we prove it.

@N3buchadnezzar Sorry man, I couldn't find a mistake.

N3buchadnezzar

@PeterTamaroff =)

There is a sign error though tries to fix

Peter Tamaroff

@Chris'ssis Note the above is the same as $$\sqrt n \int\limits_0^{\delta \sqrt n } {{e^{ - \frac{{{x^2}}}{2}}}\sin\sqrt n xdx} $$

So we can simply consider finding $$\mathop {\lim }\limits_{A \to \infty } A\int\limits_0^{\delta A} {{e^{ - \frac{{{x^2}}}{2}}}\sin Axdx} $$

First, we can get rid of the $\delta$, by writing $$A\int\limits_0^{\delta A} {{e^{ - \frac{{{x^2}}}{2}}}\sin Axdx} = A\int\limits_0^A {{e^{ - \frac{{{x^2}}}{2}}}\sin Axdx} + A\int\limits_A^{\delta A} {{e^{ - \frac{{{x^2}}}{2}}}\sin Axdx} $$

And note that $$A\int\limits_A^{\delta A} {{e^{ - \frac{{{x^2}}}{2}}}\sin Axdx} = {A^2}\int\limits_1^\delta {{e^{ - \frac{{{A^2}{u^2}}}{2}}}\sin {A^2}udu} $$

And it should be clear this goes to $0$ as $A\to\infty$.

Thus we're looking at $$\mathop {\lim }\limits_{A \to \infty } A\int\limits_0^A {{e^{ - \frac{{{x^2}}}{2}}}\sin Axdx} $$

Peter Tamaroff

7:17 PM

Minor change: we should be using $$ - \frac{{{x^2}}}{2} - \varepsilon < \log \cos x < - \frac{{{x^2}}}{2}$$ since the $+\epsilon$ fucks things up.

robjohn

@MartinSleziak: the question was undeleted

Alizter

Is lambda calculus still being studied or was it just computability proof?

Arkamis

Well, they started computing it but they're not sure when they'll stop

Alizter

7:35 PM

oops

I was trying to find a symbol

Chris's sis

7:55 PM

@PeterTamaroff OK, thanks!

Peter Tamaroff

@Chris'ssis Tell me when you prove the last limit is $1$.

Chris's sis

@PeterTamaroff OK. I'm working on a different approach now.

Peter Tamaroff

I'm leaving, but could you share it here? I'll look at it when I come back.

Chris's sis

@PeterTamaroff I'm going to do something very crazy, very very crazy. I'll let you know.

@PeterTamaroff but divine at the same time.

brb

N3buchadnezzar

8:11 PM

@Chris'ssis Heya

woo! I fixed my mistake in my formula, silly old bum.

Chris's sis

brb (I talk to my mom)

robjohn

8:32 PM

@Chris'ssis The answer was $1$, right?

Chris's sis

@robjohn back. Yes.

robjohn

@Chris'ssis okay :-)

I got it to $$\lim_{n\to\infty}\frac{n}{2^{n+1}}\sum_{k=0}^n\binom{n}{k}\frac{1-(-1)^k}{k}$$ and that I can show via the CLT to be $1$

Chris's sis

@robjohn you mean Cesaro theorem?

robjohn

@Chris'ssis Nope, Central Limit Theorem :-)

Chris's sis

@robjohn oh, nice.

robjohn

8:40 PM

I can probably do it with more elementary tools.

Chris's sis

@robjohn :)))))))) That's what I like to read, hear!

N3buchadnezzar

@Chris'ssis ^^ Did you see the question I answered, fun one sortofish..

Chris's sis

@N3buchadnezzar what question?

N3buchadnezzar

here.. :-)

Chris's sis

@N3buchadnezzar Interesting at a first glance (+1). I'll check the details a bit later.

N3buchadnezzar

8:44 PM

I also liked the similar question which relates to Elliptic functions and AMG

Chris's sis

@N3buchadnezzar I'd encourage you to provide with more answers. :-)

N3buchadnezzar

Only if I am struck by a glimpse of enlightment, I am still too dumb to do anything of value ;)

Landen

Chris's sis

@robjohn $$\int_{0}^{\pi/2} \cos^n(x) \sin(n x) \ dx=\frac{1}{2^{n+1}}\sum_{k=1}^{n} \frac{2^k}{k}$$

@robjohn then $$\lim_{n\to\infty}n \int_{0}^{\pi/2} \cos^n(x) \sin(n x) \ dx = \lim_{n\to\infty}\frac{n}{2^{n+1}}\sum_{k=1}^{n} \frac{2^k}{k}$$

The last limit may be nicely computed by Cesaro-Stolz theorem and the answer is $1$.

Transcript for

$Mathematics$ Mathematics