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15:01
@Faheem if it's related to statistics at all, maybe the people over on the stats stackexchange can help?
@DanielRust its just simple math, the linear relationship between two variables can be expressed as y=a*x, and the cubic relationship as y=b*x^3, where a=Ymax/Xmax and b=Ymax/Xmax^3
@DanielRust I am having difficulty in implementing cubic relationship
@DanielRust I have written down the formula, I just need little look over on it to know if they are correct
@Faheem Sorry i'm not familiar with terms such as 'cubic profile'.
15:32
hehe, I created so much stuff today ... let me share something new ...
$$\int_0^{\pi/2} \cos^{2013} x \sin(2017x) \ dx$$
15:57
However, this version is tough.
While walking in the park, I realozed that
The Binomial Theorem says
$$
\begin{align}
1+a_{n+1}
&=\left(1+2013a_n\right)^{1/2013}\\
&=1+a_n-1006a_n^2+O\left(a_n^3\right)\tag{1}
\end{align}
$$
Bernoulli's Theorem guarantees that $a_{n+1}\lt a_n$. Since $a_n\gt0$, $a_n$ must have a limit which satisfies
$$
(1+x)^{2013}=1+2013x\tag{2}
$$
Bernoulli's Theorem says that for $x\gt0$, the left side of $(2)$ is greater, so the only non-negative solution is $x=0$. Thus,
$$
\lim_{n\to\infty}a_n=0\tag{3}
$$
@robjohn beautiful :D
@Chris'ssis So the limit is $\frac1{1006}$
@robjohn exactly!
16:25
@Chris'ssis Hmm... I did something like this for a question not too long ago.
@robjohn hehe, yeah. This integral has a nice form $$\int_0^{\pi/2} \cos^{n} x \sin(nx) \ dx=\frac{1}{2^{n+1}} \sum_{k=1}^{n} \frac{2^k}{k}$$
16:42
@Chris'ssis Here is what I answered.
@robjohn yeah, I know that proof. Nice. I also let a comment there when you posted it.
Away for 20 min.
17:17
@Chris'ssis Just curious--how do you create such interesting problems? Do you just write out something you think looks interesting/difficult to solve, and then try to find the answer, or do you take a basic problem and add more to it until it becomes interesting?
@TedShifrin I am now, how can I help you?
Hi Charlie!
:)
@anorton hi norty!!!
@anorton how are you?
Doing ok... I have a cold, but it's tolerable... :\
How are you?
@anorton I'm fine, despite the crazy weather, that is driving me nuts
17:31
@Charlie Ah yeah... The weather can be very pesky at times.
@anorton yeah >.<
@Charlie I just noticed that you seem to have taken a break from asking/answering questions...
@anorton well noticed
I only mention it because I have taken a bit of a break from participating on the site (at least, am doing so in a much reduced state)...
Interesting
I thought about posting a question days ago, but gave up. It was too silly.
I have no sum.or integrals to share, no analysis, so no questions to ask.
17:37
Hmmm...
Are you still playing with math, though? :)
(I hope so...)
@anorton yeah, I think...
That's good...
Well, I've got to do some work on a pesky English paper, so I've got to go...
Nice talking with you! :)
@anorton Sometimes these things simply come to mind with no effort, but I'm definitely sure that the experience makes me think in a creative way, in some cases I modify some problems I meet and then I try to solve them that way, maybe sometimes I study some specific area like Euler sums and then during the research I meet some interesting things that make me ask myself different questions, or I merely dream some questions that is the funny way of coming up with new stuff. :-)
@anorton The creation process is in a way like playing football, after a while you nicely handle the ball with no effort.
@Charlie The compulsion on integrals! You're affected by it too.
@GustavoBandeira no, absolutely not.
@anorton it was my.pleasure, see you :)
17:43
@Charlie There is a bijective mapping between your avatar and you. :P
I guess you are your avatar.
@GustavoBandeira yes. Why do you tgink I chose it?
Grumpy cat understands me
@Charlie This reminds me of you.
@anorton When I attend math I never think of solving things (everybody solves things), but I try to look at the beauty of the questions and solving the problem is just a consequence of thinking this way.
@GustavoBandeira I have laugh attacks every time I watch.it
@Charlie I confess I imagine you going to university in the same fashion of that video.
17:47
@GustavoBandeira "quer votar pra mesa de students" "man, I don't vote on it, I THREW YOUR VOTE ON THE GROUND"
@Charlie Vinícius Rodrigues comes at me and say that he uses Seda Shampoo in his hair. MAN! I THREW IT ON THE GROUND! I'M NOT GONNA LET YOU DESTROY MY HAIR, I AINT GONNA BE PART OF THE SYSTEM
@GustavoBandeira hehe
@cyberskull
@Charlie
Hi, how are you?
17:54
@cyberskull fine andya?
=D CCCCCCCCCCCCCCCCCCCOMBO BRAKER.
-_-
Tenia que ser el smartpantalones
el smartpantalones ? :P
Hi @Peter
Scary that two of the starred tidbits on the right of the page are quotes from me ...
@TedShifrin 120, 120, 120.
Oh, for that problem :) Yes. How many proofs do you have? :)
And can you construct the point (other than with physics)?
18:02
@TedShifrin Well, that answer is physic-made, yes. Which is cool, but not that cool.
Right, so when you're bored in an hour or two you can try to find math proofs. I already told you to think about the multivariable calculus one. :) I give you a hint, though, for the other problem: 120 is not the answer :D
@TedShifrin él me llama de sillypantalones
You done with your tennis lessons?
@TedShifrin No lessons today. =)
@Charlie: Wish I could talk Spanish, but I can't. French, yes. A bit of German, yes. And a bit of English. Did @Peter give you the sillypantalones name?
18:03
I sometimes cover for a friend on Sundays, but not always.
@TedShifrin Aw, yiss. I did.
Ah, so you need to go get exercise, @Peter. I had three horrible long deuce games on my serve which I ultimately lost because of an excessive number of double faults (6 total) and forehand errors. I was mad at myself. But since these are the only games my opponent won, I can't be too upset.
@TedShifrin Ah! Good!
@Peter: So it remains to give a classical Euclidean geometry construction of the point, too. ... Well, I would lose to anyone younger and decent at tennis at this point; this is a colleague I've been playing for 30+ years and we're both getting old. And my game always beat his, so it's really not much challenge :(
@TedShifrin Heh, I guess the nice thing about tennis is it is also a game one plays against oneself, its really about self improvement, like math.
@TedShifrin so do I :)
18:08
Well, someone will have to teach me Spanish. A lot of my students speak it, but not natively.
@TedShifrin If you know French, you should get along well with spanish, right?
French and Latin. I can understand some Spanish, but that didn't help Spaniards in Madrid understand my French :P
@TedShifrin I am not really experienced in Euclidean constructions!
@TedShifrin I found so cute you and pedro getting along so well
@Peter: She seems a bit like a modern-day Edith Piaf (from the 60s).
LOL @Charlie: Why is that?
Seems he gets along with everybody ...
@Peter: Can you construct angle bisectors and perpendicular bisectors and equilateral triangles, etc., using compass and straightedge?
18:11
@TedShifrin Yes, sure.
And I bet you could figure out how to draw parallel and perpendicular lines, copy angles, lengths, etc. So you know as much as I do :P
@TedShifrin except me }:)
and me
LOL, I hadn't observed that. Give him more good math problems :P
18:13
Qu'est-ce que ça veut dire, @Pedro?
@TedShifrin Uses Google. Interpret it!
Either stepping back in petrified fear or saying "basta" :D
@TedShifrin The correct answer is: "Watch out, we got a badass over here."
Oh, that was not my interpretation. Which one is the badass, then? :D
@TedShifrin @cyberskull @Charlie
18:17
Ah, see, I don't know much around here.
so much to learn :P
So I learn badasses and you learn math @cyberskull?
@PeterTamaroff mira, maloculo HAHAHA
@Charlie LOL
Sure, @Jean-FrancoisRossignol. How far have you gotten?
Did you try writing out an example with $A$ a $2\times 2$ matrix?
You should always try to learn from understanding examples.
@Charlie who's bad?
@cyberskull ;)
My students think I'm pretty bad ... in several ways :D
18:24
@TedShifrin are they right?
@TedShifrin What do you mean?
Depends on your choice of meaning of "bad" :P
@TedShifrin hmmm....
If we had Robjohn, Ted and Brian Scott all here present, Pedro would go wild!
LOL ... Brian Scott drives me nuts. I don't want to deal with him.
@TedShifrin good
18:27
He is so concise.
Brian rules
@TedShifrin Ah?
@Charlie Why?
I don't like his posting complete solutions for everybody. We would have a fistfight about what learning/teaching are, I think.
@TedShifrin Hehehe, right.
I have very strong opinions about what good teaching is.
18:28
He has a different approach. Sometimes he overdoes it, dunno.
Imagine ... there, I'm bad :P
@TedShifrin What was the criticism?
I think there was a discussion on meta, and he was on the side of giving students answer books.
Maybe I'm misremembering and doing him a disservice.
But I don't think so.
It does depend on the student, no?
18:30
why not?
It just takes enormous patience and time to be a good teacher when students don't want to be good students.
Hi. I'd like to show that every positive semidefinite function/matrix is Hermitian. So thinking about eigenvalues, eigenvectors I get $x^*Ax=x^* \lambda x=x^*A^*x$ which, I think, means that eigenvectors and eigenvalues of $A$ and $A^*$ coincide. If that is correct, does it mean that these matrices coincide too? I can't find an answer to it..
Which reminds me, did @Jean-François ever answer my question?
@TedShifrin Nope.
frustré
18:31
@Julius I don't think that is enough, at least in the general case.
@Julius: What is your definition of positive semidefinite?
I don't know, there are too many professor twho don't teach, don't help, do noting, only expects students know stuff. I don't know what they think to teach is
Every book I know assumes it's symmetric/hermitian.
@Charlie: There are LOTS of bad teachers. Including ones who give beautiful lectures but don't teach. But those are rare, too.
Huh? @Peter
I'm totally baffled by @Julius's question.
18:34
@TedShifrin yes....
Which question, @cyber (now that you removed your comment)?
LOL@Peter ... I think we should retag immediately.
@TedShifrin it wasn't important, sorry
Speaking of algebraic geometry, @Peter, where are the four skew lines ? :D
@TedShifrin Ugh, darn. I am doing the Lindelöf Regular --> Normal now.
@Julius, are you going to answer my question? What is your definition of positive (semi)definite? And where did you get your equation with $x$, $A$, and $A^*$?
18:38
I have to show that every positive semidefinite complex function (i.e. $sum_{i,j=1}^n a_i\overline{a_j}r(t_i-t_j)\geq 0$, $a\in\mathbb{C^n}$) is Hermitian. So I thought that I could think about it in terms of matrices where e.g. $a_{13}=f(t_1-t_3)$
LOL, ok, @Peter.
@Julius: What's $r$? In linear algebra, to talk about a positive (semi)definite matrix, it MUST be hermitian/real symmetric to start with.
I just tried to use $Ax=\lambda x$ and multiplied by $x^*$ to get a thing that I know something about :)
That should be $f$
Ah, and what are the $t_i$?
$t\in\mathbb{Z}^n$ is in one of examples
And what does it mean to say a complex function is hermitian?
As I say, I'm baffled. I'm not picking on you. I'm just puzzled by terminology that's different from anything I've seen/done in my life.
18:42
An amazing question I received from my former teacher ... (he challenged me)
Prove that
$$\frac{F_{2n}^2}{F_{n-1}F_n}\le\binom{2n}{n}, \space n\ge1$$
$F_n$ - Fibonacci numbers
@Chris'ssis Induction?
@Chris'ssis: I haven't seen this, but there are all sorts of amazing interconnections between Fibonacci and Pascal's Triangle.
@PeterTamaroff it could be.
It says that a function is called Hermitian if $f(x)=\overline{f(-x)}$
@TedShifrin aesthetically it looks great. :-)
18:44
domain/range, @Julius?
en.wikipedia.org/wiki/Hermitian_function like this. In the example that I have $f:\mathbb{Z}\to\mathbb{C}$. Now I'm thinking that maybe it was totally wrong to think about Hermitian matrices here..
So far the only examples I can think of are maps $f\colon\Bbb R\to i\Bbb R$ given by odd polynomials, like $f(x) = ix$, $ix^3$, etc.
Hi @KaziarafatAhmed I like your avatar, very chiseled :-)
I am reading about spectral analysis, complex covariance functions, long term memory models and that's the exercise that I found at the end of the chapter.. So this is not really related to the main topic but I'm also puzzled about it :)
Well, @Julius, I see why you're tempted. If you define $b_{jk}$ by $f(t_j-t_k)$, then with the definition of a hermitian function you're giving, $f(t_k-t_j) = \overline{f(t_j-t_k)}$, so the matrix will be hermitian. And then the sum is $\sum b_{jk}a_j\bar a_k = \langle Ba,a\rangle$.
So, yes, with this definition of the matrix $B$, it is hermitian to start with because of your definition of hermitian function.
But this matrix is not naturally associated with the function $f$. We get different matrices for every choice of $t$.
@Peter: I love that French singer. I might have to buy this :P
18:54
@PeterTamaroff: Hi
@TedShifrin ;)
@FernandoMartin Sup.
Not much. I was looking for bookbinding videos.
Where did you learn how to do it?
Interesting .. I have a friend here in town (an older woman) who has been doing bookbinding for decades.
@TedShifrin, so there is no other way to show this other than by definition?
hi @FernandoMartin.
18:56
Hi @TedShifrin
@Julius: By definition a matrix is positive semidefinite iff it is hermitian and $\langle Ax,x\rangle \ge 0$ for all $x$. Then, by the Spectral Theorem, one proves that this is equivalent to requiring that all its eigenvalues (REAL by Hermitian) are $\ge 0$.
OK, guys, I need to go work on a problem a colleague gave me and prepare my graduate class. Have fun without me :P
@FernandoMartin Heh, I would *hardly*¨call it book binding. I just staple the pages in booklets, then put them in a press machine and glue the back of the book, let it dry nd that is it. No strings or whatever. It might very well fall apart in some time.
See ya, @FernandoMartin, @Peter, @Julius.
Thanks and bye, @TedShifrin
18:59
Publishers have long since quit using sutures, @Peter. That's why my books all fall apart. Makes me furious.
See ya, @cyber
@TedShifrin Ugh. Drats. I re binded my copy of Apostol. See ya!
See you, @TedShifrin
@PeterTamaroff: How do you staple the booklets so that the staples are parallel to the fold?
hi guys how is it going what's up how are you how are you doing how are things ?
@FernandoMartin If I have a stapler that gets open, I place the page on some pile of soft paper (tissue say) with a carton base and just push it, then fold the tips. If not, you have to fold the page diagonally a little (I think I told you how in rosario) to get space.
Yes, I remember you mentioned something about tissue but I didn't really understand
Ok, that makes sense. Thanks!
I have a bunch of ringed books I bought at the copier and it's a pain to know which is which
19:06
hi @JayeshBadwaik
@FernandoMartin Ah.
19:25
Each time I bring up this chat window, I feel Peter's starred pain.
@robjohn LOL, why?
@PeterTamaroff It's big and blue and there every time I look at the page.
@robjohn Ah, I have my paged zoomed in, so I don't see it.
Here is another cute calculus question: Prove that $$\frac{2(e^x-1)}{e^x+1}\le \int_{-x}^{x} \frac{\sqrt{e^t}}{e^t+1} \ dt \le x, \quad \forall x \ge 0$$
@Chris'ssis Derivatives, why not?
19:41
@PeterTamaroff yeah, it could be.
@Chris'ssis Why do you find it cute, dawg?
I'll add this question to my mega calculus problems collection.
@PeterTamaroff it's meant to be amazingly nice.
@Chris'ssis ?
@PeterTamaroff it's amazing to see how powerful $AM-GM-HM$ inequality is here.
@PeterTamaroff I gave it to some students and no one did it so far.
It seems troublesome.
@Chris'ssis You're a professor!
19:48
@PeterTamaroff No!
Aw, yiss.
@what'sup hello
@PeterTamaroff looking at these things I'm completely amazed. It's the way these math things act on me.
@Chris'ssis $$\frac{2e^t}{(e^t+1)^2}\le\frac{e^{t/2}}{e^t+1}\le\frac12$$
@Chris'ssis how are you again ? :-)
@robjohn good
:-)
19:51
@what'sup integrate over $[-x,x]$
i noted that :-)
@robjohn we may also write things as$$\sqrt{\frac{e^t}{(e^t+1)^2}}$$
@Chris'ssis what's the difference ?
@Chris'ssis well that is how you would show the left hand inequality
@robjohn right
19:53
oh ! got it .
@what'sup this is the difference $$\sqrt{\frac{e^t}{e^t+1}\cdot \frac{1}{e^t+1} }$$
ok i got it
@what'sup then you apply the inequality mentioned above.
!!!!!!!!!!!!!!!!!!!!!!!!!!!!
i knew that
@Peter: Now I'm really pissed off ... :)
19:57
@TedShifrin Oh, noes, why?
did someone try to solve Fibonacci numbers question without induction ?
That person who didn't respond to my question had put a plea to julien on his question. I put a comment, saying I was annoyed he hadn't responded to me, and reiterated that he should try a $2\times 2$ example. (The question is immediate, btw.) I reiterated that one learns to understand math by doing/understanding examples. Not only did he not respond, he got the mods to remove my comment. And he's again asking "won't anyone help him"? GRR ... Never talking to him again.
@Chris'ssis You don't even need that... since $\frac{e^{t/2}}{e^t+1}\le\frac12$, multiply both sides by the left side.
Maybe I need a sabbatical from here.
As regards the inequalities, this is one of the most beautiful inequalities I've ever seen (I don't give you the link to upvote me, but only to see it) - math.stackexchange.com/questions/150770/…
@robjohn right
19:59
@TedShifrin Don't do that... once people fall into the abyss of reality, we never see them again!
LOL ... Very funny, @robjohn.
I'm getting crankier and crankier about most of the people who are here, @robjohn.

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