Mathematics - 2014-06-09 (page 2 of 3)

Bananarama

3:07 PM

does this look ok to you? I hope I didn't make a mistake math.stackexchange.com/a/828041/33907

Balarka Sen

@Alyosha Sure.

N3buchadnezzar

@Chris'ssis you gave the original one without limits

Balarka Sen

It's called quotient category.

N3buchadnezzar

@skullpatrol no brain, no pain

Chris's sis

@N3buchadnezzar Yes, but we wanted to compute a certain definite integral.

Balarka Sen

3:13 PM

@N3buchadnezzar no strain, no brain

N3buchadnezzar

:p

@Chris'ssis eith what limits?

Chris's sis

@N3buchadnezzar $$\int_{1}^\infty \frac{\log t}{t(t-1)(\alpha t - 1)} \mathrm{d}t$$

N3buchadnezzar

the integral blows up at $\beta$

Balarka Sen

@N3buchadnezzar $\beta < 1$

Chris's sis

@N3buchadnezzar It's related to the integral problem I proposed yesterday. Prove that $$\int_0^{\infty} \frac{x}{(e^x-1)(e^{x+\log\left(\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}\right)}-1)} \ dx=\frac{\displaystyle \zeta(2)-\phi\left(\frac{3}{5}\zeta(2)-\log^2(\phi)\right)}{\phi-1}$$

N3buchadnezzar

3:24 PM

what is alpha?

greater than one, less than one ?

mick

hello world

How to prove to a girl that she belongs with you ? :)

I posted a new question today

if anyone cares

Bananarama

@mick I don't think this is the right chat for that

mick

@Bananarama It was meant to get a reaction and it did.

And btw there is no right chat for that

but I still want a good answer :)

the answer is probably : girls do not get proofs :p

Bananarama

@mick There are girls that are awesome in math

mick

Yeah but they fail to understand the proof that they belong to me :)

@Bananarama

N3buchadnezzar

3:38 PM

Nobody got chat for that

Bananarama

oh, yes girls don't seem to understand that with me either.

mick

hehe

Maybe someone wants to see my question of the day ...

Bananarama

They seem to understand even less if you explain it to multiple girls at once

mick

link or so

Bananarama

@mick why do you have that profile pic?

mick

3:40 PM

yeah but at some point they magically believe it with some guy who did not even prove it ... @Bananarama

nabla blah

@Banana It's a picture of himself, duh

Bananarama

perhaps that might have something to do with people not getting your proofs then

mick

yeah its a picture of me :) But I had to put chuck norris face on it make it more impressive ...

nabla blah

Good choice @mick

mick

thanks @nablablah

MrWho

3:42 PM

Hi everybody

mick

Maybe I should post my question here with a link ?

MrWho

Does anyone know anything about the principle of nested rectangles?

@BalarkaSen ^

@BalarkaSen books.google.com/…

mick

I wrote an intro to my profile ...

0

Q: Question about kth root of a reduced ring element.

Let $n > 1$ be a positive integer. Let $k > 1$ be a positive integer. Define the reduced rings $f_n = \Bbb R[X_n]/(1+(X_n)^{n})$ How do we know if $(X_n)^{1/k}$ is an element of $f_n$ ? Possibly trivially related : How do we know if $(-1)^{1/k}$ is an element of $f_n$ ? For instance I wonder if...

the question

G.T.R

@Chris'ssis math.stackexchange.com/questions/828090/…

MrWho

@mick You 15?are you kidding me?

Bananarama

3:48 PM

@MrWho balarka was born yesterday

and he knows more math than me

MrWho

@Bananarama I'm absolutely bamboozled by your dubious proposition!

Bananarama

he's 14

MrWho

Interestingly however, I'm not embarrassed by the math knowledge he has got by that age but the shaky fundamental he's building his math upon.

2

mick

Yes im 15 and Balarka is 14.

MrWho

@mick @BalarkaSen If you both are 15 and 14 then I'm -20 :)

@BalarkaSen Tell me about the principles of nested rectangles?!

Balarka Sen

3:53 PM

wassgoingon?

@MrWho What's that?

mick

Well MrWho people at university are often 20 yo so why would that be strange ...
@everyone
@MrWho

@MrWho who ? what ?

Balarka Sen

@MrWho Too shaky. Might fall down any time.

MrWho

The principle of nested rectangles is used to clarify the uniqueness of a particular pointed in which contains the whole nested rectangles in a complex plane, this is what I've got from it.

Balarka Sen

@MrWho I don't know the name. tell me what's the statement.

Spelling big names is not mathematics.

MrWho

@BalarkaSen Yeah, having shaky fundamentals in math would result in crappy conseuqences!

Chris's sis

3:56 PM

@G.T.R math.stackexchange.com/questions/828090/…

mick

@BalarkaSen chuck norris !

MrWho

@BalarkaSen Check out the books.google.com/…

Balarka Sen

@MrWho Well, I have done the necessary prerequisites for what I am reading right now.

Will read more fundamental when I need more stuffs.

@MrWho Ah.

MrWho

@BalarkaSen I know, but I'm not convicting you of not having information about my question.

I'm simply asking about the principle, I didn't get it, the explanation was not really good.

And also I'm reading the translation of the book right now !

Balarka Sen

@MrWho never read that stuff. but seems easy to prove.

MrWho

3:58 PM

In my own language, so the it leads to much more vagueness I suppose.

What do you mean by "never read that stuff"?

it has been introduced in the first chapter as a prerequisite for the most of the other chapters too!if I don't learn them I'll stuck somewhere.

Balarka Sen

the book I have don't have that theorem in.

MrWho

@BalarkaSen WHat's your book?

Balarka Sen

@MrWho Lipshitz-Spiegel.

MrWho

@BalarkaSen Well, this is not a Analysis book, it has skipped lots of cool stuff too!

G.T.R

@Chris'ssis I'm afraid people will downvote if you don't elaborate :P

Balarka Sen

4:01 PM

@MrWho Huh? Aren't you reading a complex analysis book?

MrWho

@BalarkaSen Yeah

@BalarkaSen I skipped real Analysis.

Balarka Sen

Lipshitz-Spiegel is a complex analysis book too.

And it doesn't have that stuff as a prerequisite.

So it's just an exercise for me.

mick

Complex analysis rules !

Balarka Sen

@mick Yeah. Real analysis sucks though.

Bananarama

real analysis is awesome

mick

4:06 PM

What do you guys think of my profile ?

Balarka Sen

@Bananarama Not really.

Bananarama

it's grotesque

Balarka Sen

it's the analysis of really ugly functions.

Bananarama

@BalarkaSen isn't it a matter of taste?

Balarka Sen

but oh, well, i need fourier one way or another.

@Bananarama didn't you get the pun?

Bananarama

4:07 PM

no, I'm sort of bad at getting jokes

Balarka Sen

really

real analysis

Geddit?

MrWho

@BalarkaSen It shouldn't be a good book, because it doesn't even cover the nested rectangles principle, principles are important(!), why? because we're building our ideas upon them and if we haven't been aware of such principles then we're doing math blindly in the wicked world of blindness.

mick

the problem with complex analysis is usually that there are many strong conditions to a theorem ... like being analytic on a half-plane or such ..

but when you have many singularities you seem to get stuck often ...

right ?

MrWho

@mick right!

Balarka Sen

Not every book has same perspective, @MrWho

MrWho

4:09 PM

@mick Anyway, complex analysis has its own break through(s) in which makes it cool!

mick

@MrWho that was a bit unexpected ... I expected defenses of complex analysis.

MrWho

@BalarkaSen And those books are not really math books!

Balarka Sen

@MrWho I like complex analysis for number theory's sake =D

mick

@MrWho I only know analytic number theory ? any others ?

Balarka Sen

@MrWho Well, each to his own.

Chris's sis

4:10 PM

@G.T.R No, I won't do that since it's too easy. How about this one $$\lim_{n\to\infty} \frac{\displaystyle \int _a^b f^{n+1}(x) dx}{\displaystyle \int _a^b f^{n}(x) dx}$$ ?

Balarka Sen

@mick Me a little.

mick

@BalarkaSen Like what ?

MrWho

@mick Well, have you read the analytic number theory book written by Apostol?

Balarka Sen

I know prime number theorem and stuffs and reading sieves right now. I am also interested in elliptic curves and modular forms.

but my algebraic NT is pretty weak. gonna learn in again.

@MrWho it's a very good book. but not for beginners.

mick

@MrWho I mean what other breakthrough does complex analysis have apart from number theory and perhaps complex dynamics ?

@BalarkaSen

Balarka Sen

4:13 PM

I don't know.

mick

I dont believe it has any :)

thats a bold statement i guess

MrWho

@mick It's not a bold statement, it's a dubious proposition at best!

Balarka Sen

I think I should get back to reading sieves.

mick

or answer my question @BalarkaSen

Daniel Fischer

This is a bold statement!

Balarka Sen

4:15 PM

@mick what question?

mick

@DanielFischer no thats a silly message :)

@BalarkaSen the link I gave.

Balarka Sen

@mick You know field theory, right?

Bananarama

@DanielFischer this source says the opposite

mick

@BalarkaSen some , not much ... why ?

Balarka Sen

@mick $F[x]/(f(x)) \cong F(\alpha)$ for some irred $f(x)$ in $F[x]$ with root $\alpha$

MrWho

4:17 PM

@BalarkaSen Would you often solve the problems provided after reading each chapter? sometimes I enjoy reading the ideas and methods rather than getting the feeling of applying them instantly to the let's say silly and unreal problems designed to be solved to gain the skill in that particular area!

Balarka Sen

@MrWho I do it if it looks cool and hard and I leave it if it looks made-up.

MrWho

@BalarkaSen Well, can you rewrite $tan(6x)$ in the form of $tan(x)$ using the Euler formula?

Chris's sis

@G.T.R better try this version (a little bit funnier) $$\lim_{n\to\infty} \frac{\displaystyle \int _a^b f^{n+2}(x) dx}{\displaystyle \int _a^b f^{n}(x) dx}$$ @BalarkaSen @N3buchadnezzar don't miss it - under the conditions from here math.stackexchange.com/questions/828090/…

Balarka Sen

No need for Euler's formula.

MrWho

@BalarkaSen The question is insisting on using Euler identity!

Balarka Sen

4:20 PM

@MrWho Then use it.

No one's stopping you.

There is a much weaker De'Moivre to do that for you though, but essentially it's a consequence of Euler's formula.

MrWho

@BalarkaSen However, my math engine has been shut down these days, I feel burned-out!

@Chris'ssis Give me some motivations - you source of motivation :D

Balarka Sen

@MrWho Well, I find it an obvious consequence of Euler's formula.

$\mathcal{R} \; (\exp(i\theta))^6 = \mathcal{R} \; \exp(i6\theta) $ and $\mathcal{I} \; (\exp(i\theta))^6 = \mathcal{I} \; \exp(i6\theta) $

mick

@BalarkaSen Obvious ? sine and cosine would be Obvious. I wonder how you do the then tan case.

Balarka Sen

@mick $\sin/\cos = \tan$

mick

@BalarkaSen yes , so ?

Balarka Sen

4:25 PM

$\sin(6\theta)/\cos(6\theta) = \tan(6\theta)$

mick

ah wait you use the complex conjugate in euler's formula !

right

?

MrWho

@BalarkaSen Yeah, I know the sine and cosine, they're represented by the binomial distribution.

Balarka Sen

@MrWho I don't know what binomial distribution is.

MrWho

@BalarkaSen :O

@BalarkaSen Then get back to the high school math :)

mick

@BalarkaSen you arent going to write out sin(6x) , then cos(6x) and then simplify , are you ?? must be a more elegant way.

Balarka Sen

4:27 PM

@MrWho What is it about?

mick

@MrWho distribution = statistics ? you mean binomial theorem ?

Chris's sis

@G.T.R $$ \lim_{n\to\infty} \frac{\displaystyle \int _{1/2}^{1} \operatorname{Li}_2(x)^{n+2} dx}{\displaystyle \int _{1/2}^{1} \operatorname {Li}_2(x)^{n} dx}=\zeta(2)^2$$

Balarka Sen

yeah, binomial theorem is it? @MrWho

MrWho

@BalarkaSen Oh, sorry, yeah

Binomial theorem I meant.

Balarka Sen

@MrWho Stop spelling silly and big names and show me the math instead.

I don't recall half the name of the stuffs I did.

MrWho

4:29 PM

@BalarkaSen Binomial theorem is a big name for you?!

Balarka Sen

@MrWho No, but binomial distribution is.

mick

lol

MrWho

It's quite obvious from the name itself what I'm talking about!

Balarka Sen

@MrWho No, it's not.

Hypergeometric function and hypergeometric distributions are not the same thing.

mick

Who will be first to give a nice proof of this tan(6x) thing with euler's ?

Balarka Sen

4:31 PM

@Chris'ssis interesting. How did you get that?

MrWho

In addition, binomial distribution is not a big name too! it's just a simple probability distribution in discrete math to calculate the chance of success after limited number of attempts.

Balarka Sen

@mick not me.

@MrWho i ain't know no probability.

sorry.

MrWho

@BalarkaSen That's okay, if I made you confused, just forgive me :)

Balarka Sen

@MrWho That's fine, if I made you embarrassed for my shaky fundamental math, forgive me.

mick

@MrWho actually you made the mistake by naming incorrectly ... thats no issue , until you add that someone should study more or again or such ...

Why is Balarka's math shaky ??
incomplete i think

MrWho

4:33 PM

No problem, c ya later all :)

mick

@MrWho show us the elegant tan(6x) proof plz ...

using euler !

Chris's sis

@BalarkaSen @G.T.R $$\lim_{n\to\infty} \frac{\displaystyle \int _{1/2}^{1} \operatorname{Li}_2(x)^{n+2014} dx}{\displaystyle \int _{1/2}^{1} \operatorname {Li}_2(x)^{n} dx}=\zeta(2)^{2014}$$

Balarka Sen

@Chris'ssis Whoa. That's pretty elegant.

Can you do the same for, say, $\zeta(3)$?

Chris's sis

@BalarkaSen Yes.

@BalarkaSen $$\lim_{n\to\infty} \frac{\displaystyle \int _{1/2}^{1} \operatorname{Li}_3(x)^{n+2014} dx}{\displaystyle \int _{1/2}^{1} \operatorname {Li}_3(x)^{n} dx}=\zeta(3)^{2014}$$

mick

Anyone here willing to look at my question ( link i gave ) ?

btw how do you like my profile ? I added an intro tekst.

0

Q: Question about kth root of a reduced ring element.

Let $n > 1$ be a positive integer. Let $k > 1$ be a positive integer. Define the reduced rings $f_n = \Bbb R[X_n]/(1+(X_n)^{n})$ How do we know if $(X_n)^{1/k}$ is an element of $f_n$ ? Possibly trivially related : How do we know if $(-1)^{1/k}$ is an element of $f_n$ ? For instance I wonder if...

polynomials ring-theory divisibility roots-of-unity

Show your skills !! :D

Student

4:48 PM

Hello. Does anyone know if the answer to my latest question here is trivial like my previous two questions?

I'm getting a bit desperate because I really don't get it.

I only got 10 views but I can't put a bounty on it yet.

mick

@Student I think you will get an answer anyway ...

Student

It's not looking too promising at the moment.

I wonder why there is a 2 day waiting period for bounties.

mick

me too

I was hoping to get more attention to my question by coming to chat.

Failed today ...

So unless someone wants to talk to me , im going ...

Bananarama

chat isn't for attracting attention to questions, most people answer the questions they can answer

mick

@Bananarama It takes a lot of time to get an answer ... its a bit frustrating ...

In the beginning I hoped a higher rep would get more attention to my questions and more/better answers.
Guess not :/

Who wants to talk ??

Bananarama

5:03 PM

gaaaaaah why won't people accept my answer!!

mick

where

Bananarama

1

A: formula for number triangles

A recursive solution: number the rows $0$ to $r-1$ with $0$ being the fattest, notice that the difference between consecutive numbers in row $k$ is $2^k$. Denote by $f(n)$ the number in the last row of a triangle with $n$ rows. Then $f(n)=2f(n-1)+2^{n-2}$ Notice $f(1)=0,f(2)=1,f(3)=4$ From th...

Balarka Sen

Bleh. I don't understand this.

Anyone familiar with sieves?

mick

no

srr

bye all

N3buchadnezzar

meh

r9m

5:19 PM

@Chris'ssis So you are active on the main !!! :D

Chris's sis

@r9m For a second ... :-) Have you seen the last stuff I created?

r9m

5:30 PM

@Chris'ssis ah sorry .. I was afk ..

ya I have :D

Mike Miller

Good question, @G.T.R

r9m

5:48 PM

@Chris'ssis have you seen this ? :) .. looks interesting .. do you think the solution could be simpler than what Prof. Kouba provided there ?

G.T.R

@r9m do you know the answer when we throw the condition $f\geq 0$ away ?

r9m

@G.T.R ya .. last night I missed that condition and answered it wrong .. $7/3$

Chris's sis

@r9m yeah, I think it can be done easier.

G.T.R

@r9m lol Kouba(at first) and I answered as well forgetting about it :P

r9m

@G.T.R irk .. hehe .. the problem is simpler without that condition ..

@Chris'ssis would you add your answer there .. ? (now that you are active finally) :)

Chris's sis

5:55 PM

@r9m Well, I'm not active in the real sense. :-)

r9m

@Chris'ssis okay .. and you are answering like 'Cleo' ? one liners .. :P thats not fair :P lol

Chris's sis

@r9m hehe, that meant "without giving many explanations". ;)

r9m

@Chris'ssis no that means "without giving 'any' explanation" :P lol

bolt from blue :P

Chris's sis

@r9m :-)))))))))))

r9m

@Chris'ssis I missed your 7 papers .. can you link me the urls of those images ? ;) .. I found even after one removes the image from the chat .. it is still there in the image hosting site .. if one types the correct url .. the image pops up without a hitch .. :|

@Chris'ssis I get it .. thanx :D

Chris's sis

6:04 PM

@r9m Welcome ;)

r9m

@Chris'ssis Will you propose it to some cool math magazine problem section ? (Its possibly too tough for any limited time competition) :D

Chris's sis

@r9m No. I'd like to add it to my book one day ... (in case I'd be able and lucky enough to publish a book with questions created by me).

r9m

@Chris'ssis Well I am an eager fella waitin for that "Book" .. I guess so is @G.T.R ;) :D

@Chris'ssis hmm .. you did something there !! :O ... it will possible take me years before I can begin to understand the sorcery conjured in those 8 paper .. some Dark Awesome magic :P :D

bbl .. dinner time

Chris's sis

6:19 PM

@r9m That proof can be optimized to a certain extent. I'll try these days to get a nicer form of it, a shorter one. I saw some points where I can reduce the calculations a lot, especially in the last parts.

G.T.R

6:50 PM

@Chris'ssis does $\sum _{k=1}^\infty \sin \left(\sqrt{k}\right)$ converge ? What about $\sum _{k=1}^\infty \sin \left(\sqrt{k}\right)/k$

@Chris'ssis the partial sums of the first one look beautiful when plotted

Chris's sis

@G.T.R Well, as regards the first one things are clear, aren't they? It diverges.

G.T.R

@Chris'ssis I want proofs :P

Chris's sis

@G.T.R Proof? hmmm

MrWho

@mick Well, I know the proof, but I'm not in the sense of typing it :)

@mick Don't have patience to deal with latex stuff !

G.T.R

@Chris'ssis yeah, it doesn't go to $0$ for the first one. But it does for the other, which looks highly convergent

Chris's sis

7:05 PM

@G.T.R Do you think we can use what Noam Elkies did for this series here? math.stackexchange.com/questions/630890/…

G.T.R

@Chris'ssis I'd like to, but it seems to converge; very slowly but still

MrWho

I hate inequality, it just pisses the whole certainty I got about what I'm thinking!

Chris's sis

@G.T.R We might try this gun ... en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula

G.T.R

@MrWHO if you hate them, they'll hate you back

MrWho

@G.T.R Yeah :)

G.T.R

7:19 PM

@chris'ssis I'll ask on main

Balarka Sen

7:48 PM

@G.T.R Have you tried Abel test?

OK, nevermind.

Chris's sis

8:01 PM

@robjohn That series is proudly part of my Series Hall of Fame. I'm so happy for that achievement! I should give it my name. :-)

Surely, there are some points in my proof that can be subject to the optimization process.

Balarka Sen

@Chris'ssis Like?

Chris's sis

@BalarkaSen For instance, I have in mind an integral that I tackled by beta function, digamma function and other stuff, but this one I could have done by other tricks, much easier.

Balarka Sen

8:16 PM

I am trying to optimize bits of my proofs too.

robjohn

@Chris'ssis Nice. I haven't had a chance to look at it.

Chris's sis

@robjohn Not even a bit? :-(

robjohn

@Chris'ssis not beyond looking at the problem statement... I've been busy with work this morning

Chris's sis

@robjohn OK

Balarka Sen

Can you let me see it, @Chris'ssis?

Chris's sis

8:18 PM

@robjohn It seemed to me so different from all I've seen before in terms of series.

Balarka Sen

OK.

Ah, so the definite integral you gave us was a part of this solutions. Nice.

Chris's sis

@BalarkaSen Indeed. :-)

Balarka Sen

@Chris'ssis So the problems you do weren't as off-the-context as I thought it was.

Chris's sis

@BalarkaSen No, not at all. All I post is something related to things I work on.

Balarka Sen

@Chris'ssis But you won't tell us the context, even though you know it. How meanie.

(looking at page 3)

Chris's sis

8:22 PM

@BalarkaSen :-)))

brb

Balarka Sen

Interesting. That's interesting, @Chris'ssis.

Chris's sis

@BalarkaSen Thanks. Glad you like it. :-)

G.T.R

@Chris'ssis math.stackexchange.com/a/828354/66096

Chris's sis

@G.T.R Nice. However, I remain at my first thought, Euler–Maclaurin formula.

G.T.R

@Chris'ssis post it then

Balarka Sen

8:32 PM

@Chris'ssis Bombing the mosquito.

Chris's sis

@BalarkaSen That is a nuclear weapon. :-))))))

Chris's sis

8:53 PM

@G.T.R Somehow, you reach the same work as Davide. (with Euler–Maclaurin formula)

Jasper Loy

9:13 PM

It is rather quiet in chat today. Also, no lhf in sight.

nabla blah

hi

Jasper Loy

@nablablah What are you studying these days?

nabla blah

9:29 PM

@JasperLoy Analysis

Jasper Loy

@nablablah From which book?

nabla blah

PMA

Jasper Loy

@nablablah That is one of my nine holy books.

Chris's sis

@robjohn the double series in the fifth part can be done in a far nicer way than what I did in that proof.

Ted Shifrin

9:54 PM

heya @Jasper mr eyeglasses @Chris'ssis

Jasper Loy

@TedShifrin Hey about to retire professor, lol.

Chris's sis

@TedShifrin Hello! How are you doing, professor? :-)

Ted Shifrin

slaps @Jasper

just fine, thanks, @Chris'ssis, and you?

nabla blah

lol Still calling me "mr eyeglasses"

Old habits

Ted Shifrin

well, you kept them in your name

Jasper Loy

9:55 PM

@TedShifrin I am surprised nobody has ever flagged your slaps, it seems, lol.

N3buchadnezzar

meh

m.e.h

Ted Shifrin

I'm sure you want to flag, @Jasper

Chris's sis

@TedShifrin I'm optimizing some proofs here. :-)

Ted Shifrin

make sure some compactness argument proves there's an optimal one, @Chris'ssis

Jasper Loy

@TedShifrin There are two things I don't do in chat: flag, star.

Chris's sis

9:56 PM

@TedShifrin :-)

Ted Shifrin

weird: heya @N3 - You can perform this action again in 1 seconds - retry / cancel

nabla blah

I thought I was impatient

Ted Shifrin

aren't you, @nabla?

nabla blah

yes

Ted Shifrin

math isn't good for impatience

N3buchadnezzar

9:58 PM

I saw edge of tomorrow half an hour ago

impatience leads to frustration, frustration leads to wrong answers, and wrong answers leads to the dark side

Transcript for

$Mathematics$ Mathematics