Mathematics - 2014-12-15 (page 6 of 10)

Venus

7:00 PM

@Huy You kidding right?

Huy

@Venus: No, and two of my classmates witnessed it.

They were only able to look at him though.

Daniel Fischer

@Venus I'm more old-school, he and Becks were too obsessed with their appearance for my liking. But, Ronaldo was/is a truly outstanding player.

Venus

@Huy Prove it! No picture = hoax :D

Huy

@Venus: I didn't take a picture with me bumping into him, obviously.

robjohn

@MathyPerson No... either click on the link I gave above, or click on the $\LaTeX$ in chat link. Either will take you to the installation page.

Venus

7:02 PM

@DanielFischer I only see both Becks & CR7 because they're handsome. I've never seen them playing

Huy

@Venus: There were girls standing all around him hoping to get an autograph or a picture with him and didn't. I just accidently bumped into him.

Mathy Person

@robjohn: I was referring to the star as the place where I add bookmarks

robjohn

@MathyPerson Yes, the entry has a star with 22 next to it

@MathyPerson the link takes you to this page: math.ucla.edu/~robjohn/math/mathjax.html

Hippalectryon

@robjohn Let $a\in\mathbb{R}$, Is there a matrix $A$ such that $\exp(A)=\begin{bmatrix}1&a\\a&1\end{bmatrix}$ ?

Hakim

@Chris'ssis Did you ever encounter this limit?
$$\lim\limits_{n\to\infty}\dfrac n{\ln n}\left\{\dfrac1\pi\left(\sin\left(\dfrac\pi{\sqrt{n^2+1}}\right)+\sin\left(\dfrac\pi{\sqrt{n^2+2}}\right)+\cdots+\sin\left(\dfrac\pi{\sqrt{n^2+n}}\right)\right)^n-\dfrac1{\sqrt[n]{e}}\right\}.$$

skullpatrol

7:05 PM

@Hippalectryon can't you see he's busy right now :P

Hippalectryon

@skullpatrol I thought it would be easy for him

@skullpatrol Well, you can always answer :D

Chris's sis

@Hakim No. It looks very nice.

Huy

@Hippalectryon: The determinant of that matrix is $1 - a^2$, so it is $0$ for $a=\pm 1$ and thus one of the eigenvalues has to be zero. So, no.

Venus

@Huy You should have acted like these fans :D

Jasper Loy

I just answered a lhf.

robjohn

7:06 PM

@Hippalectryon I don't think so, but let me try something.

Hippalectryon

@Huy What if $a\neq\pm1$ ?

Mathy Person

@robjohn: I do see that page, but for some reason I can't bookmark the "Start ChatJax" section. I can only bookmark the installation page

Hakim

@Chris'ssis Do you know how to approach it?

Huy

@Hippalectryon: You can take the log iff the eigenvalues are strictly positive.

robjohn

@MathyPerson can you click and drag the "start ChatJax" link to your bookmark bar?

Hippalectryon

7:07 PM

@Huy Oh I haven't seen the matrice's $\ln$ yet

Huy

@Hippalectryon: The matrix is symmetric, so you can diagonalise it. The exponential of a diagonal matrix is just the diagonal matrix with the exponential of its entries and you can compute the matrix logarithm for a diagonalisable matrix in the same way.

Jasper Loy

Mod nominations close in 50 minutes, quick!

Mike Miller

@robjohn I know this is locked because of the contest rules, but can you remove the algebraic geometry tag?

Huy

@Venus: I was a bit perplex, to be honest. I didn't immediately recognise him, I wasn't wearing my glasses. I realised it after his bodyguards dragged me away. :D

Hippalectryon

@Huy Hmm but how does the log help ?

Mathy Person

7:10 PM

@robjohn: I tried that, but it's not adding it for some reason

Chris's sis

@Hakim First, how would you compute $$\lim_{n\to\infty} \left(\sin\left(\dfrac\pi{\sqrt{n^2+1}} \right) + \sin\left(\dfrac\pi{\sqrt{n^2+2}}\right) + \cdots+\sin\left( \dfrac\pi {\sqrt{n^2+n}}\right)\right)$$?

2

Huy

@Hippalectryon: If $A$ is symmetric, you can write $A = T D T^{-1}$ for some diagonal matrix $D$. Then, $B = \exp(A) = T \exp(D) T^{-1}$, where $\exp(D) = \operatorname{diag}(e^{d_1}, e^{d_2}, \dots)$. You can take the logarithm of matrices in the same way.

Venus

I wanna get Handegg hat so badly, could someone here help me to vote this answer until +7? Haha

Hippalectryon

@Huy I do understand that, but I don't see how that helps solving the problem

Venus

@Huy He has bodyguards? No wonder, he's one of the richest athlete in the world

Chris's sis

7:13 PM

@Venus the limit I posted above in one line?

N3buchadnezzar

@JasperLoy Nominate Santa, I think he has been a good boy this year.

r9m

@Venus (+1) just for hats sake =P :D

Huy

@Hippalectryon: If $B=e^A = \begin{pmatrix} 1&a\\a&1 \end{pmatrix}$, since it is symmetric, you can diagonalise and then take the log of the eigenvalues.

Venus

@Chris'ssis It 's trivial. $$\Huge\mathbb{Q.E.D.}$$

3

Chris's sis

@Venus LOL

Huy

7:15 PM

@Venus: He was attending the award of the Ballon d'Or.

Hippalectryon

Too many stars

Venus

@r9m Śukriyā :D

@Huy What's that?

Hippalectryon

@Huy Well I understand that $\log(B)=A$, but knowing the existence does not tell me how to actually find the actual matrix, does it ?

@Chris'ssis But we can't pin you to the starred board :D

3

Chris's sis

@Hippalectryon :-)))))))))

Venus

3 more upvotes for this one :D

Mathy Person

7:16 PM

Can someone help me with this? $For any positive integer n, let G(n) be the number of pairs of adjacent bits in the binary representation of n which are different. For example, G(10)=3 because the bits of 1010_2 change at all three places and G(12)=1 because the bits of 1100_2 change only from the fours to the twos place.

For how many positive integers n<2^{10} is it true that G(n)=2?$

Huy

@Hippalectryon: I told you how to find the actual matrix, I think.

Venus

I wanna have Handegg hat so badly :D

r9m

@Venus hahahaha :D !! where are you from ? (if you do not mind me asking)

Venus

@r9m Sarawak, Malaysia

Huy

@Hippalectryon: You diagonalise $B = T D T^{-1}$, then you take the log of the diagonal entries of $D$ and then you multiply by $T$ and $T^{-1}$ from left respectively right.

Hippalectryon

7:17 PM

@Huy I don't really know how to diagonalize that matrix. For that I'd need to find both eigenvectors...

Oh wait me stupid, I know the eigenvalues

r9m

@Venus ah ! :-)

Venus

@r9m You?

r9m

@Venus West Bengal, India :)

robjohn

@Hippalectryon $$\frac12\begin{bmatrix}\log\left(1-a^2\right) & \log\left(\frac{1+a}{1-a}\right)\\ \log\left(\frac{1+a}{1-a}\right) & \log\left(1-a^2\right)\end{bmatrix}$$

Venus

@r9m I see it on Google. Where's Taj Mahal located?

r9m

7:19 PM

@Venus Agra ! its about 1277 km away from my home ! (I have been to Taj Mahal once ... but I was too young to remember anything now)

Venus

@r9m I know, I mean is that far from West Bengal? Have you been there before?

Balarka Sen

It is far @Venus

Jasper Loy

Wow, I got 35 points for my lhf, lol.

Venus

@r9m It must be a romantic place. @BalarkaSen It's even farther than my place

2 more upvotes for this answer guys. For the sake of Handegg hat :D

Hippalectryon

@robjohn Are you sure ? I get $$M=\begin{bmatrix}-1&1\\ 1&1\end{bmatrix}\cdot\begin{bmatrix}1-a&0\\ 0&1+a\end{bmatrix}\cdot\begin{bmatrix}-1/2&1/2\\ 1/2&1/2\end{bmatrix}$$

r9m

7:24 PM

@Venus Idk ! maybe it is .. with the right company :-)

Jasper Loy

@Venus Wow, @venus is a genius just like @Chris'ssis, lol.

Venus

@JasperLoy Thanks for the upvote Jasper

Balarka Sen

@Venus Made out of the blood of 10,000 workers... sure is romantic.

Venus

@PranavArora Hi! Have you seen this answer? :-)

Pranav Arora

@Venus

Yes

robjohn

7:26 PM

@Hippalectryon You need to have a space at least every 80 characters or chat inserts nasty HTML spaces that mess the LaTeX up.

Pranav Arora

Saw it now :P

Venus

@BalarkaSen Well, it's certainly a bad story but it's history.

robjohn

@Hippalectryon I fixed the previous one

Hippalectryon

Thanks

@robjohn Anyway, isn't that right ?

Venus

@PranavArora Thank you so much! It's 7 already and I pass my daily caps for the first time :D

Pranav Arora

7:28 PM

:)

robjohn

What is $M$ supposed to be?

Jasper Loy

@Venus You get a mortarboard badge for getting 200.

Hippalectryon

@robjohn That leads me to $\begin{bmatrix}-\ln(1-a)&\ln(1+a)\\\ln(1-a)&\ln(1+a)\end{bmatrix}$

@robjohn $M$ is $A$, sorry

Balarka Sen

@Venus I don't find a burial romantic from any respect.

Venus

@JasperLoy I didn't notice that I've already earned that badge :D

robjohn

7:30 PM

@Hippalectryon I have verified the matrix with Mathematica. Exponentiating it gives your initial matrix

Venus

@BalarkaSen OK, that's your opinion & I respect that.

Hippalectryon

@robjohn Ah thanks. So, it works for all $|a|<1$ ?

Hakim

@Chris'ssis I have no idea, maybe will have to use a probabilistic argument, no?

Chris's sis

It seems today is a great day to me, I'm terribly creative!:-)

robjohn

@Hippalectryon yes

Chris's sis

7:32 PM

@Hakim Note each term of the form $$\sin\left(\dfrac\pi{\sqrt{n^2+k}} \right)$$ can be approximated by $$\dfrac\pi{\sqrt{n^2+k}} $$ where the latter terms can be nicely squeezed and we're done (when considering $n$ large enough).

Venus

I look like Iron Woman using Handegg hat. Cool! ^^

Thanks everyone for your help. Now, I have 8 hats :D

Balarka Sen

I only have three.

Venus

@BalarkaSen I can help you to get more. I can start by downvoting your question :D

Balarka Sen

Fortunately, I have asked no questions.

And thanks, but no thanks.

Venus

I have asked @Chris'ssis to do so to earn Business in the front, Party in the back hat

robjohn

7:37 PM

@MathyPerson Sorry for the timeout there... have you tried right-clicking and adding it that way?

Venus

@Chris'ssis Need help for that? You just call me :-)

I wish I could donate some of my hats to you

robjohn

@Hippalectryon did you map back to the original space?

Hippalectryon

@robjohn I found my error, thanks.

Now i'm working on a new exercise :D

Chris's sis

@Venus Maybe it's not bad for a while. ;)

robjohn

@Hippalectryon Oh, good. You just missed the last step, right?

Hippalectryon

7:39 PM

@robjohn Indeed

Jorge Fernández

how do you make bullet points look good?

MathGod

Hello everyone! Can anyone please suggest a software for writing latex in microsoft word?

Jasper Loy

@MathGod What do you mean by latex in word?

evinda

@DanielFischer Do we have to give as a parameter to the function partialQuicksort the position q+1 ?

Venus

@robjohn I have accepted your answer. You should change your current hat with a new one after getting a silver badge (Enlightened)

robjohn

7:40 PM

@MathGod You mean for images or something like what LaTeX does?

Jorge Fernández

writing latex in microsoft word is sacrilege.

r9m

@Hakim @Chris'ssis I think @robjohn has evaluated the limit somewhere ! lemme look for it .. also $\displaystyle \sum\limits_{k=1}^{n} \sin \dfrac{1}{\sqrt{n^2 + k}} = \pi -\frac{\pi}{4n} - \frac{3\pi+4\pi^3}{24n^2} +\mathcal{O}(\frac{1}{n^3})$

Jasper Loy

@JorgeFernández I still don't understand the question.

Chris's sis

@r9m that one is elementary because of the magnitude of the denominator that place everything in infinity area. All you need to do is to see that for $n$ large enough we have that $$1-\epsilon \le \sin \dfrac{1}{\sqrt{n^2 + k}}/\left(\dfrac{1}{\sqrt{n^2 + k}}\right)\le1+\epsilon $$

Jorge Fernández

me neither, but it sounds bad

MathGod

7:41 PM

@robjohn I meant how to write equations in microsoft word. I have to write a paper and then convert everything to pdf.

robjohn

@Venus I only have two hats and the other looks terrible

r9m

@Chris'ssis I saw Hakim's Q above .. :) I'm looking for @robjohn's answer to a similar question ! (I'm not answering it)

Hippalectryon

I haz da russian hat :3

Jasper Loy

@MathGod Very simple. Use the Equation Editor. It should come with Word, just look for it and bring up the whole toolbar.

robjohn

@r9m where are you looking for an answer?

Balarka Sen

7:42 PM

@Hippalectryon LEL

how do you get that secret hat?

robjohn

@JasperLoy That just produces images of the equations, doesn't it?

Venus

@robjohn You'll earn Enlightened badge, afterwards you'll get a new hat

Hippalectryon

@BalarkaSen No idea it's a "secret hat". Maybe @MikeMiller or @robjohn knows

Daniel Fischer

@evinda Not sure about the context. q+1 is the lower end of the array range passed in one of the recursive calls (if that is not skipped).

Jasper Loy

@robjohn Well, it produces a WYSIWYG document.

Chris's sis

7:43 PM

@r9m or one may simply use the known inequality $\sin(x)\le x \le \tan(x)\space x\in[0,\pi/2)$

r9m

@robjohn you answered a similar question months ago .. I think I read it .. so I'm searching for it in your list of answers :)

Venus

Is there a hat that looks like a Princess' crown?

Jasper Loy

@robjohn When I was teaching in school, all I had to use was Equation Editor in Word for equations and Shapes in Word for graphics. Word is very powerful. I still prefer it to LaTeX, lol.

r9m

@Hakim here :)

Hippalectryon

@BalarkaSen Oh i think I know :)

evinda

7:44 PM

@DanielFischer Oh yes, right!!! How can we then find the position that we have to save? :/

robjohn

@r9m the $\lim\limits_{x\to0}\frac{\sin(x)}{x}=1$?

Balarka Sen

@Hippalectryon ah?

Chris's sis

@r9m By squeeze theorem, we are done with $$\lim_{n\to\infty}\sum_{k=1}^n \dfrac{1}{\sqrt{n^2 + k}}$$ Q.E.D.

r9m

@robjohn I linked it to Hakim above ^

Daniel Fischer

@evinda Sorry, which position are you talking about?

Hippalectryon

7:45 PM

@BalarkaSen The title says it all

robjohn

@JasperLoy but as I remember the equation editor only puts out limited resolution images. Not terribly good for priinting

r9m

@Chris'ssis I think last time we encountered it I killed it in 3 ways -_- (I'm really not looking for an explanation here ..)

MathGod

@robjohn Ain't there a software that enables us to write Tex codes in microsoft word that can appear as equations?

Balarka Sen

@Hippalectryon i still don't get it

robjohn

@MathGod Not that I know of, but I am not a Word power user

Chris's sis

7:46 PM

@r9m hehe, OK :-) I have an amazing question for you

Venus

@Hippalectryon It looks like an old communist hat

3

Balarka Sen

oh

Chris's sis

@r9m do you see this one? math.stackexchange.com/questions/1069376/… Can we find a proof in one line? Is that possible? :D

Hippalectryon

@Venus I'm wearing it :D

Hakim

@r9m Oh, thanks! :-)

evinda

7:47 PM

@DanielFischer We want to find the interval to which the p closest elements to the median element belong... Don't we find it from the recursive calls? Or am I wrong? :/

robjohn

@Hippalectryon or be the first to upvote an answer (perhaps on your own question)

Jasper Loy

@robjohn Well, the resolution on screen and on paper are both fine to me. Just as good as LaTeX.

Balarka Sen

@Hippalectryon

LEL

Hippalectryon

@robjohn I think it appeared for me after going to the review queue

Venus

@Hippalectryon I think mine cooler

r9m

7:47 PM

@Chris'ssis I read both answers ! :) no new ideas atm though ! :)

Hippalectryon

@Venus I think that picture is someone else's >.>

robjohn

@JasperLoy Nah... LaTeX will print at full resolution. Nicely typeset. Word did not look so good.

Hippalectryon

@BalarkaSen That looks like something from pastafarism :3

Jasper Loy

@MathGod I suggest you use LaTeX or Word and not both.

Venus

@Hippalectryon I'll change my ava later

Hippalectryon

7:48 PM

@Venus You don't have to :)

Do whatever you want

Daniel Fischer

@evinda We computed the interval before we started the partial sorting. During the partial sorting, we move the elements that belong to that interval into the right places.

Balarka Sen

someone wants the Sumo Judge badge again

Venus

Anyway, the primary votes is about to start. Who must I vote other than @DanielFischer-sensei?

Balarka Sen

@Hippalectryon LOL!

Jasper Loy

@Venus I am voting for Daniel, Jyrki and Pedro.

Balarka Sen

7:49 PM

Same here @Jasper

Hippalectryon

I'm voting for the dude with the lowest rep :3

Jasper Loy

@BalarkaSen Fools seldom differ.

Balarka Sen

Haha

Venus

@Hippalectryon To be honest, I'm a beautiful girl ^^

Hippalectryon

Don't ask me e__e

Jasper Loy

7:50 PM

@Venus Please put up your picture then!

r9m

@Venus pic please ! ;-)

Huy

Guys, please, this is a chatroom about maths, not a dating website, @JasperLoy, @r9m.

4

Balarka Sen

no pic, thanks.

Venus

@JasperLoy @r9m Please be patient. I have to make up first :D

Hippalectryon

@Huy Thanks

Jasper Loy

7:51 PM

@Huy When did you become Ted?

3

Balarka Sen

LEL

evinda

@DanielFischer That's what I haven't understood... How can we compute the interval before starting the partial sorting?

Venus

@Huy LOL, you have my star for that

Hippalectryon

Too many starred messages >.>

r9m

@Huy LOL :P

Venus

7:52 PM

Where's Ted? I miss him so badly ^^

Hippalectryon

@Venus He's cooking infants

MathGod

Does anyone know where to vote?

Hippalectryon

He's getting an Infant Cooker 2000 for Christmas

Balarka Sen

@Hippa "Look round, the Frenchman loves its blaze" that's meant for you

Huy

@Hippalectryon: I got a microwave for Christmas.

Daniel Fischer

7:53 PM

@evinda We know the place where the median element goes to when sorting. And we know how many elements we want. These elements must be adjacent to the median after sorting. So we can compute the positions.

Hippalectryon

@BalarkaSen >.>

Venus

@robjohn Can I vote the mods in separate days? For example: today I vote @DanielFischer-sensei, then tomorrow I vote another one, etc

@Hippalectryon Yuck!

MathGod

@Venus Where do you vote?

Hippalectryon

@MathGod Voting begins in 4 mnutes

Huy

@Venus: I need four more reviews for my awesome football hat.

Venus