Mathematics - 2012-02-20 (page 2 of 3)

t.b.

16:01

It was okay... I was sort of fearing lots of complaints about my latest answer (I committed the terrible sin of mixing real and complex :/) and I think there are some slight inaccuracies, but I'm too lazy to fix right now.

Kannappan Sampath

@tb I see.

t.b.

How's yours?

Dylan Moreland

@KannappanSampath It seems like TeX SE worked out rather well.

Kannappan Sampath

I have had a good day. I have been reading up on Linear Algebra for my mid semester exams....

Matt N.

Ello : )

Il y a

16:03

@tb good afternoon

Kannappan Sampath

@DylanMoreland Yes. I had now fixed the error.

t.b.

Hi Matt, Il y a and Dylan

Kannappan Sampath

Matt, how's your PDO?

Dylan Moreland

Hi all.

Matt N.

Hi Dylan.

Dylan Moreland

16:04

Good to see you around, @tb.

Matt N.

@KannappanSampath Not so well, I'm quite tired. But working on it. : )

Kannappan Sampath

@MattN You will have to catch up as Friday is four days nearer!

t.b.

@DylanMoreland It's been a while, yes. How have you been?

@MattN coffee injection?

Gigili

Matt N.

@KannappanSampath Yeah yeah, plenty of time... : )

Dylan Moreland

16:05

I'm alright. Woke up late; need to catch the train down to see Matt E.

Need to remind myself of what Griffiths transversality is.

t.b.

@Gigili not really :)

ello anyway...

Matt N.

@tb I'm not hardcore enough. : ) I just finished a big cup of maccha latte, my recently discovered secret weapon. Now I'm waiting for it to kick in...

Daniil

Solving mundane automata-theory exercises is actually more fun than I thought/

Matt N.

First. : )

Gigili

Ello @tb, @MattN, @Daniil, @DylanMoreland and etc.

Matt N.

16:07

Ello Gigili. : )

Kannappan Sampath

I see ello so many times and whatnot.

t.b.

@DylanMoreland Oh, I seem to remember that I once heard the term in a summer school on Shimura varieties. But that was a loooooong time ago :) Good luck!

Kannappan Sampath

Ello all :-)

t.b.

@MattN I see... Hope it works :)

Dylan Moreland

@tb Shimura varieties, exactly!

Daniil

16:12

@Gigili Hi :D

Il y a

@MattN maybe, waiting is not the best strategy? )

Matt N.

@Ilya : P

Kannappan Sampath

What in India is as effective as Coffee injections @tb?

Dylan Moreland

But it comes from Griffiths' work on Hodge theory. And probably he was trying to prove the Hodge conjecture because that's what he was always doing.

Il y a

@KannappanSampath is he from India?

Josué Molina

16:13

@KannappanSampath What will I see if I scan your avatar with my phone?

Kannappan Sampath

@Ilya But he might know of substitutes in India, you see.

@JosuéMolina You're likely to see something about Langland's programme.

Dylan Moreland

Somehow I think it's a condition on whether your Hodge number "jumps" can come from geometry. Like, "is there a moduli space for these things".

Il y a

@Kan indeed, why not

t.b.

@DylanMoreland well, I went there mainly because I had to. But there was a very nice course on symmetric spaces by Eberlein and the big thing was that Ullmo and Yafaev presented their ideas on proving the Oort conjecture. But all that was way beyond my head.

@KannappanSampath I don't know. Maybe four liters of chai?

Kannappan Sampath

@tb And my stomach takes the shape of balloon which will burst as quickly as in an hour and I'll die!

t.b.

16:17

well that's the minor downside of it :)

Kannappan Sampath

minor downside!!!

Il y a

I have two excellent books an Stochastic processes in continuous time

minor downside. as you like, @tb

Josué Molina

underage downside?

t.b.

@Ilya :)

Il y a

and I have to solve for an optimum: which one to read first

ah, hell! there is the 3rd one (((

and all of them are excellent

Josué Molina

16:22

@Ilya Are you Russian?

t.b.

@Ilya what are you talking about?

Il y a

@JosuéMolina because of hell or because of blind smiles?

Josué Molina

@Ilya ((((( <- only Russians do that. :P

Il y a

@tb Revuz, Yor "Continuous Martingales and Brownian motion", Karatzas and Shreve "Brownian motion and Stochastic Calculus" and Oksendal "Stochastic Differential equations"

@MattN hey! show respect, you :)

Matt N.

: D

Kannappan Sampath

16:24

Is this a start for a Marathon removed race?

Il y a

I would also add books by Shiryaev and commit suicide

t.b.

a suicide?

Il y a

@tb mmm... suicide

Matt N.

@tb Don't -- he's touchy about being corrected.

t.b.

@Ilya without article.

Matt N.

16:25

(at least when I do it)

Kannappan Sampath

the?

Il y a

@tb thanks a lot for correcting me! (@Matt: :p)

Matt N.

@Ilya I anticipated something like this : P Predictable.

Kannappan Sampath

: )

t.b.

@Ilya Many of my friends used Karatzas and Shreve. But I suspect Revuz and Yor know what they're talking about, too :)

Il y a

16:26

@MattN that means that I am measurable w.r.t. predictable sigma-algebra. not so bad, since Brownian motion is predictable as well.

@MattN I wonder when 5 minutes will finally pass

Matt N.

@Ilya It's 2 minutes.

Il y a

@MattN phew still 1.38

Kannappan Sampath

????? scratching his head

Matt N.

@Ilya : D

Il y a

;)

Matt N.

16:29

@Ilya DUEL!!!

2

Il y a

@MattN I can't resist this challenge! )

t.b.

Oh, duels make teddy bears leave... :s Sorry guys, I have to go

Il y a

@tb hm. see you

t.b.

I'll be back soon, promised...

Josué Molina

I now understand Galois.

Matt N.

16:30

@tb Bye. Take care!

Kannappan Sampath

See you soon tb

Matt N.

</3

Il y a

@JosuéMolina don't start it. just don't start it

t.b.

Bye all, take care you too

Matt N.

@Ilya I was going to write "and I can't stop laughing" but I've stopped now.

Il y a

16:30

@MattN nice )

Matt N.

Today's visit was short.

Il y a

@MattN you know, someone may be suspicious and find a correlation between the moments teddy leaves and moments you heart breaks. I think the value of the correlation is slightly above $1$

Matt N.

@Ilya What do you mean "correlation"? It's causal.

Il y a

Probability theory tells us that there are no causalities or dependences. There are independences and correlations

in your deterministic world, the causality is just a perfect correlation

Matt N.

Going to read about Fourier series now.

Josué Molina

16:37

@Ilya And in your world?

Il y a

@JosuéMolina there are independences and correlations

Kannappan Sampath

facepalm

Il y a

@KannappanSampath be more careful with your face ) you have the only one. It would be better if you stop palming it\

user19161

16:54

@KannappanSampath Yeah don't touch your face with your palms too often or you will get acne.

Matt N.

Hi @JasperLoy. I think you are right, "so to say" is not proper English. But I'm also 90% sure that I copied it off a native speaker.

user19161

@MattN Of course, I am always right, just that you may realize it only 9000 years later. :-)

user19161

Anyway this is the link to the online CALD.

user19161

dictionary.cambridge.org

user19161

You can install its search engine in your browser too by navigating from there.

Rajesh D

16:57

Hi

Matt N.

I don't install things into my browser but thanks : )

user19161

@MattN Eh I am sure you at least use Adobe Flash?

Matt N.

Not that I'm aware of.

user19161

Hmm maybe HTML will replace Flash one day but not any time soon.

user19161

In youtube you can choose html version but some videos are not so stable I heard.

Matt N.

17:00

I didn't notice. But then again I only use youtube once in a while.

user19161

@MattN Wow, then you are super weird!

Jonas Teuwen

What browser has it? Some have it pre-installed 8-).

Matt N.

Chrome.

Jonas Teuwen

And what is installing? I making a bookmark installing?

Matt N.

Probably preinstalled.

Jonas Teuwen

17:00

Chrome has it already.

user19161

@JonasTeuwen That explains it.

user19161

I am eagerly awaiting the release of Debian Wheezy. It should be out in Dec.

user19161

It will be the first Debian release with the GNOME shell.

user19161

I think Windows 8 and Office 2012 will be out in Dec too, for Microsoft lovers.

Jonas Teuwen

GNOME Shell?

I would switch immediately.

I have Arch now.

user19161

17:06

@JonasTeuwen Arch is a rolling distro, not to my liking. And being always up to date, of course it has GNOME shell for a long time already.

Matt N.

If I compute the coefficients of my Fourier sine series of a function $f$, why can I swap the sum and the integral?

Jonas Teuwen

@JasperLoy Yes, I would switch from GNOME Shell.

@MattN Do you know Lebesgue integration?

Matt N.

@JonasTeuwen To do so, $\sum_{n=0}^\infty b_n$ would have to be finite, right? Because then I could use the Lebesgue dominated convergence theorem.

@JonasTeuwen Yes.

Jonas Teuwen

By the way, that is quite filthy, why would you do it this way?

I'd just do $f = \sum \langle e_n, f \rangle f$.

Matt N.

I don't see why it's filthy but for now I'm not worried about dirt. : )

Jonas Teuwen

17:11

Then I wouldn't worry about the switch either 8-).

Why would you need to do it?

Matt N.

I have $f(x) = \sum_n b_n \sin (\frac{\pi n x}{L})$, I multiply both sides with $\sin (\frac{\pi m x}{L})$ and then I integrate.

Jonas Teuwen

Yes, filthy.

So I wouldn't worry about the switch either.

Matt N.

$$ \int f(x) \sin (\frac{\pi m x}{L}) dx = \int \sum_n b_n \sin (\frac{\pi n x}{L}) \sin (\frac{\pi m x}{L}) dx$$

Jonas Teuwen

Yes, in what norm does that series converge?

That's all the stuff you're shoving under the rug if you do this.

Matt N.

@JonasTeuwen I don't know, I thought it converged pointwise to $f$ on $[-L,L]$.

Jonas Teuwen

17:14

It will always converge in $L^2$ for $L^2$ functions but the pointwise convergence is very hard (Carleson-Hunt theorem).

Matt N.

Why do I know that it converges in $L^2$? Should I see that?

Jonas Teuwen

Because $e_n(\theta) = \exp(i n \theta)$ is a basis for $L^2$.

Matt N.

Now it makes more sense! Thank you! : )

And how do I know which spaces have which basis? (For the next time, I mean)

Is $\sin(nx), \cos(n x)$ also a basis for $L^1$?

Jonas Teuwen

All $L^p$ spaces have that thing as a basis, that is a theorem by Riesz I believe, but then you don't have the nice expression anymore in terms of a Fourier series.

Yes, but the norm in $L^1$ can't be written as an inner product, so that is not useful here.

The good way to go is is to *define*
$$a_n = \frac1L \int_0^L f(x) \textrm{e}^{-2 \pi i n x/L} \, \textrm{d}x.$$

Matt N.

But I'm not sure I understand: a basis is a thing that lets you express things using finite linear combinations. The Fourier series above is infinite, so I can't argue that that's a basis and therefore the thing converges.

Jonas Teuwen

17:17

Your argument only gives some heuristics for what the expression for the Fourier coefficients should be.

@MattN In functional analysis we use Schauder basises for infinite dimensional spaces, not Hamel bases (which they don't have).

Matt N.

@JonasTeuwen But the other way is much more "expressive" in showing how these coefficients were derived!

Jonas Teuwen

Yes, but you shove all the technicalities under the rug.

You don't know a priori if that thing will converge.

And in what mode...

It's fine to gain some intuition.

That is why I said that it is filthy.

Matt N.

@JonasTeuwen Exactly, that's why I'm trying to understand why it does.

Jonas Teuwen

Then I wouldn't worry about switching those.

You just do so, find the coefficients and then prove that those work.

You also have such an argument to determine what would be a nice expression for the Fourier transform (let the period go to $\infty$).

Matt N.

@JonasTeuwen I'm coming to that next.

@JonasTeuwen It's like a chicken-egg-problem: To find the coefficients I need a theorem that gives me convergence to $f$ of a series I want to compute.

Jonas Teuwen

17:33

@MattN Well. You first try to find the coefficients that way and suppose everything behaves nice, then you prove that that expression works :-).

Matt N.

Yes, I know. But it's still disturbing.

Can I get anywhere if I assume that $\sum_n b_n \sin(\frac{n\pi x}{L})$ converges pointwise to $f$?

I mean with respect to swapping sum and limit.

If $f$ is continuous and I assume the series converges pointwise I can swap limit and sum because then I can apply Lebesgue's theorem for $g(x) = \sum_n b_n = f(L) \in \mathbb{R}$.

Jonas Teuwen

You could do that.

Matt N.

But then I can only do Fourier series for continuous functions.

Jonas Teuwen

...

That's why you need to take that formula and verify it.

Matt N.

What do you mean?

Jonas Teuwen

17:46

Try to prove that that formula holds in different modes.

That is what one often does in mathematics.

Matt N.

What do you mean by modes? Different norms?

Asaf Karagila

@tb Drats. I wanted to tell you something!

Matt N.

18:21

@AsafKaragila He said he was leaving because I'd said "duel!" to Ilya. You may blame it on me.

Daniil

How can I prove that a language $(ab*)*$ has a star height <https://en.wikipedia.org/wiki/Star_height> of two?

Asaf Karagila

I rather blame it on Ilya.

Matt N.

: D

N3buchadnezzar

0

Q: Proof the inequality $n! \geq 2^n$ by induction

I'm having difficulity solving an exercise in my course. The question is: Prove that $n!\geq 2^n$. We have to do this with induction. I started like this: The lowest natural number where the assumption is correct is $4$ as: $4! \geq 2^4 \iff 24 \ge 16$. The assumption is: $n! \ge 2^n$. ...

This is an exact duplicate?

Asaf Karagila

Of your mother.

Matt N.

18:27

: D

N3buchadnezzar

@AsafKaragila I have nothing to say to you landlubber

DUEL!

robjohn

Wow, yesterday was busy while I was gone to Disneyland :-)

Matt N.

@robjohn Aren't you a bit old for that? : )

(Was it fun?)

robjohn

You are never too old for Disneyland.

It was great fun. I was there with a friend, his wife and 8-year old daughter.

Asaf Karagila

I have only been to one amusement park, and it was very crummy. Disneyland sounds unreal to me.

robjohn

18:30

@AsafKaragila I grew up going to Disneyland; it is really a magical place (especially to me and my wife)

Matt N.

@robjohn Sorry about yesterday, I was being distracted.

robjohn

we both enjoy going there.

@MattN don't worry; I was gone :-)

Matt N.

@robjohn Do they let you take your dog?

Daniil

Is it true that some satan-worshipers hold their monthly meetings in the Disneyland in the core of some rolllercoaster?

Asaf Karagila

@robjohn So your growing function was discontinuous? You only grew up in Disneyland?

Jonas Teuwen

18:31

@MattN Yes, different norms.

Matt N.

@robjohn : )

robjohn

@MattN definitely not. It is very crowded at times, and there are no animals except those that the disabled need.

Asaf Karagila

@JonasTeuwen Are they equivalent?

Matt N.

@JonasTeuwen Yay! I thought you'd stopped talking to me :,)

Asaf Karagila

@MattN He was just too drunk to find the keyboard.

Jonas Teuwen

18:32

@MattN No, I was not paying attention.

@AsafKaragila No.

Asaf Karagila

@JonasTeuwen Liar liar Hardy space on fire!

robjohn

@AsafKaragila he could find the keyboard; he just didn't know what to do with it :-)

Matt N.

: D

Matt N.

18:47

Can I write every odd function as a series of sines?

N3buchadnezzar

Only finite I guess..

Jonas Teuwen

@MattN Yes.

@MattN Well, you know, $C^1$.

Matt N.

@JonasTeuwen I can write every odd $f$ that is continuous and differentiable as a series of sines?

N3buchadnezzar

If you have a matrix with a vertical line in it, what is it called?

Jonas Teuwen

Yup, with pointwise convergence. (I think integrable and continuous is enough)

N3buchadnezzar

18:53

I am trying to recreate it in latex...

Jonas Teuwen

You should mention which mode of convergence you want.

Often $L^2$ convergence is enough.

Matt N.

@JonasTeuwen I haven't thought about that. But like I wrote earlier: assuming pointwise convergence I can swap limit and integral and then compute the coefficients like that.

Jonas Teuwen

It is way easier with $L^2$.

Matt N.

Actually, why do I want $L^2$ convergence?

@JonasTeuwen I see.

Jonas Teuwen

(Cauchy-Schwarz)

Because that is easier.

Pointwise convergence of Fourier series is hard, certainly when the functions you begin with are only in $L^p$.

Asaf Karagila

18:56

Interesting. I knew that if a question has -4 it won't show on the main page, but I recall that edits and answers still bump it up there.

Matt N.

@JonasTeuwen Oh, no, I was talking about assuming pointwise convergence, not showing it : )

Jonas Teuwen

@AsafKaragila Because maybe the edit might improve the question?

Asaf Karagila

Yes, but I answered the question and it didn't get bumped; and I do not recall it was bumped - but it was indeed edited.

Matt N.

@JonasTeuwen And you're telling me to verify that $$\left \| f(x) - \sum_{n=0}^N b_n \sin(\frac{n \pi x}{L}) \right \|_{L^2} \xrightarrow{N \to \infty} 0$$

I finally understand what you're trying to tell me : )

Jonas Teuwen

Yes. Write it as an inner product.

Asaf Karagila

19:00

You should use \left\| and \right\|

Jonas Teuwen

And yes you should.

Or \bigl and \bigr or something like that.

Asaf Karagila

Or both!

Jonas Teuwen

Yeah.

Asaf Karagila

Much better.

Jonas Teuwen

If you do $_{L^2}$ I'd put in like $_{L^2(\mathbf T)}$ or otherwise just $_2$.

Matt N.

19:02

What's the $T$?

Asaf Karagila

It's short for his last name.

Jonas Teuwen

Torus.

Matt N.

What's wrong with $L^2$?

Asaf Karagila

He is trying to get $L^2$ to be named after him, the jerk.

Jonas Teuwen

There are many $L^2$'s. If you're going to be explicit be totally explicit.

I don't like "half work".

Matt N.

19:02

@JonasTeuwen Right.

Asaf Karagila

@JonasTeuwen Now that's a lie.

Jonas Teuwen

What is?

Matt N.

Now it's too late to edit.

Jonas Teuwen

Unless your underlying space is the same everywhere and you mention this somewhere.

But I'd just define \LH or something.

Matt N.

@JonasTeuwen It's $[-L, L]$.

Jonas Teuwen

19:03

Yes, the torus.

Or circle rather.

Asaf Karagila

$S^1$ for the win.

Jonas Teuwen

Doesn't matter here, does it? $\exp(- 2 \pi n i x/L)$ for $x = L$ and $x = -L$ is...?

Matt N.

Yes.

Jonas Teuwen

Good.

Victor

@robjohn - Could you explain something in your answer here, my question is on the comment under the answer : math.stackexchange.com/questions/70125/…

robjohn

19:27

@MattN periodic functions with certain smoothness conditions have convergent Fourier series. It is fairly simple to see that the cosine components are $0$.

Jonas Teuwen

Well, I was just arguing that when you talk about the convergence of Fourier series you should mention which mode 8-).

Matt N.

As Jonas pointed out to me: convergent in $L^2$ : )
Thanks, robjohn!

Jonas Teuwen

Well, now he is talking about pointwise convergence.

Matt N.

How can you tell?

Asaf Karagila

He's an analyst.

That's all he does.

Jonas Teuwen

19:40

I don't know. I just know. Maybe because the easy pointwise convergence theorems require some smoothness.

@robjohn So you don't sleep much? How do you do it? 8-). I'm even very tired after 9-10 hours.

Matt N.

@JonasTeuwen Same here.

Victor

@robjohn - May you answer my question please?

Asaf Karagila

20:00

I am going to the laundromat. Bye.

robjohn

@Victor try this

and you might want to read this

Victor

@robjohn - Thanks, but i think i can't read that and have to ask (by the way, i want to get a few solution, not just one), thanks again, bye

Jonas Teuwen

@AsafKaragila I hire a lady to do the laundry.

dilbert.com/strips/comic/1992-08-03

robjohn

@Victor what about it can you not read?

robjohn

20:24

not very polite about things.

N3buchadnezzar

@robjohn Indeed

robjohn

20:51

My kitten closed the lid on my laptop :-)

Josué Molina

Quick question: does anyone know how to use LaTeX on a Wordpress blog?

robjohn

@JosuéMolina I don't, but there are some here on Wordpress.

Matt N.

What an exhausting day.

@robjohn What's the opposite of overdose?

N3buchadnezzar

Narcose?

If I have a function f(x), and then look at f(a - x) what does this actually do? Does it mirror my function around a?

Kannappan Sampath

@JosuéMolina $latex blah$

Neal

20:56

@JosuéMolina You could try to edit the line activating MathJax directly into the header of the site's HTML file

Kannappan Sampath

Replace blah with your equations or anything you please

Jonas Teuwen

@robjohn A kitten?? :D.

Kannappan Sampath

@Neal What do you mean?

Jonas Teuwen

@N3buchadnezzar Did you try to draw a graph of I have no idea... $f(x) = x$?

N3buchadnezzar

it looks like it reflects it around the x axis, at the point a

robjohn

20:58

@JonasTeuwen yes, a non-adult cat :-)

Jonas Teuwen

@robjohn Cute! Do you have a picture?

@N3buchadnezzar Did you see what $f(-x)$ does?

Neal

@Kannappan mathjax.org/demos/use-in-web-platforms There was an article in the most recent edition of the Notices about it

N3buchadnezzar

I am trying to show that $$ \int_{a}^{b} f(x) \, \mathrm{d}x = 0$$ if $f(x)$ is symmetrical around $( a + b ) / 2 $

Kannappan Sampath

Well. The graph translates by $-a$ and is reflected about $y$ axis! @N3buchadnezzar

Jonas Teuwen

@N3buchadnezzar Set $a = -1$ and $b = 1$, can you solve it now?

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$Mathematics$ Mathematics