Mathematics - 2012-08-20 (page 7 of 7)

Jonas Teuwen

23:00

So, now I am wondering why you would say homeomorphic to instead of metric...

t.b.

@OldJohn yes, completely metrizable means that there is a complete metric inducing the topology and if you're homeomorphic to a completely metrizable space then you can simply pull that metric back via the homomorphism.

Old John

my definition of Polish space was "A Polish space is a topological space metrisable by a metric for which it
is complete and separable. "

Jonas Teuwen

I figured that having each point as a cut point actually easily gives you an order.

@OldJohn Yes, so homeomorphic to completely metrisable separable space.

t.b.

That's true. There are also order-theoretic and topological characterizations of the line.

Old John

@JonasTeuwen yes - it might not be complete in the metric it has

t.b.

23:02

You can find them relatively easily when Googling. (when I was looking for Ward's result, I stumbled over them).

Jonas Teuwen

Yes, but... but... I prefer to figure it out myself.

Only when frustration goes through the roof I will look it up.

t.b.

That's a good attitude to have.

Old John

I wrote this in my thesis: " a metric space which is not complete
may still be a Polish space, since there may be another metric which gives
rise to the same topology and for which the space is then complete."
Hope I got it right!

t.b.

As long as you don't waste too much time with figuring stuff out that you could easily look up :)

@OldJohn perfectly right.

Old John

@t.b. Great :)

t.b.

23:04

An open interval in $\mathbb{R}$, for example.

Old John

@t.b. yep

@JonasTeuwen have you got Dudleys book: Real Analysis and Probability?

Jonas Teuwen

@t.b. Well I actually figured that like big guys just copy stuff from others will the original guy was quite dumb... kinda.

@OldJohn No, but I have read it.

That's like the only thing I know a bit - measure theory and probability.

Old John

@JonasTeuwen It was one of my main references for Polish space stuff

Jonas Teuwen

What? Hmm... Do I have the good book in mind?

t.b.

@JonasTeuwen It's a matter of attitude, after all, no? Some want success, others understanding. Maybe the latter are kinda dumb. But hey, I prefer to be a kinda dumb idealist rather than a successful idiot.

7

Old John

23:09

@t.b. ditto

Jonas Teuwen

@t.b. My advisor said today: Well, you figure out many things and solve quite some nice problems but in the end you need THESIS". Also, he wants to read something. I said: "I do not like to discuss unfinished business". Hmm. But I gave something.

I just grep something together.

I don't really fancy "success". The type of success I want I will not obtain without understanding.

I have a finished result, but I dislike the proof. Not enough insight. I get slightly pushed to publish it.

t.b.

@JonasTeuwen Sure, but look, while you're writing your thesis, you're kind of safe. There's no real pressure on you to produce a lot. Enjoy that while it lasts.

Jonas Teuwen

I sometimes feel lots of pressure, but I bet it is just egosyntonic.

t.b.

Well, you should try to write up something halfway publishable for that. You can still polish it for the thesis itself. Try not to have too high standars. High standards are good, but perfectionism can be counterproductive in this world...

BenjaLim

Hey guys

Jonas I integrated by parts

Jonas Teuwen

23:14

Yes, that is what my advisor said. Something like, I understand, but you need to publish to be able to get a job later on.

@BenjaLim Yeah. I know. You pinged me like 14 times about it.

Son of a crack.

BenjaLim

@JonasTeuwen I am having some trouble trying to get an estimate for the Dirichlet Kernel

Jonas Teuwen

What estimate do you want?

t.b.

@JonasTeuwen Unfortunately this is true. Mostly.

BenjaLim

@JonasTeuwen Here

t.b.

@JonasTeuwen I filed that under lost cause :)

BenjaLim

23:15

That sine term in the sum is ugly

Jonas Teuwen

@BenjaLim What does $f \ast D_N$ denote?

BenjaLim

convolution

Jonas Teuwen

@t.b. 8-).

No bloody hell, what is it.

BenjaLim

@JonasTeuwen One definition would be the one I gave in terms of the partial sums of the fourier series

Jonas Teuwen

The $N$-th partial sum.

BenjaLim

23:16

@JonasTeuwen yes.

@JonasTeuwen The sine term is ugly.

Jonas Teuwen

So you want that in the middle of the interval plus some stuff $\pi/N$ (where you have odd constant so that it works for all numbers).

BenjaLim

well, if little $n$ is odd the term in the sum is zero.

Jonas Teuwen

You know what the series will converge to right?

Is this about the Gibbs phenomenon?

BenjaLim

No.

Jonas Teuwen

It will converge to $\frac12$.

BenjaLim

23:18

It's just some random thing that guy posted

Jonas Teuwen

So now you are looking right next to that point and he asks you to quantify the Gibbs phenomenon.

BenjaLim

Uh,,,,,

Jonas Teuwen

Yeah, you take step functions, because that way you can quantify it for all jump discontinuities.

Pierre gave you this? Good job.

BenjaLim

No it's a math.se question

Jonas Teuwen

Oh, right missed the author.

BenjaLim

23:19

some guy posted this and I'm trying to answer it.

@JonasTeuwen How do I estimate the sum??

Jonas Teuwen

I would first shift the whole thing.

BenjaLim

to?

I spent close to 4 hours last night on this

Jonas Teuwen

You are required to find a lower bound on an alternating sum, you know how to do that right?

Also, it is about 0.08.

t.b.

Well, Ben is a bloody algebraist, don't expect too much :)

BenjaLim

No.

t.b.

23:23

Sorry...

BenjaLim

@t.b. It feels a lot better to be able to write down these estimates :D

In AT there is a lot of black magic happening

t.b.

The black magic will feel a lot less magical shortly

BenjaLim

@t.b. Last night you should have seen the integrals I crunched out. I forgot I knew how to integrate by parts.

t.b.

That's not good.

BenjaLim

@t.b. Well at least Van Kampen does not feel like that anymore. I used it like 50 million times the past 2 weeks

t.b.

23:24

People used to get a Fields medal for integrating by parts...

Jonas Teuwen

I can write down the solution easily, or do you want me to help you...?

t.b.

Don't give it away, Jonas.

BenjaLim

@JonasTeuwen I don't want you to give it away.

I feel I am close to the solution.

t.b.

Today I unpacked my old keyboard from the mid-nineties. It's a lot of fun typing with it.

BenjaLim

I spent a lot of time last night trying to get bounds for the input of the sine function in the sum

@t.b. it makes a lot of sound

t.b.

23:26

Yes, and it's a bit slower, but I'll get used to it again.

BenjaLim

@JonasTeuwen why is the sum alternating?

Jonas Teuwen

Maybe I missed something.

BenjaLim

The only way I see it is changing sign is from the sine terms.

Jonas Teuwen

What you do is just compute $\lim_N S_N(f)(\text{mess})$.

BenjaLim

you have $(1- (-1)^n)$ being either zero or 2

Dilate to infinity?

How does that hellp?

Jonas Teuwen

23:27

Yes, all positive so it grows.

BenjaLim

no wait a minute

the sine terms surely some of them are negative

Jonas Teuwen

Right, you have

BenjaLim

@JonasTeuwen Oh DA FUQ

The input into sine

Jonas Teuwen

$$S_N(f)(\pi + \frac\pi{N}) = \sum_{N} a_n \sin(n (\pi + \frac\pi{N}))$$

I probably messed up. Let me see and eat and Scotch.

BenjaLim

@JonasTeuwen Yes you're right they're all positive

$N + 1/2$

Jonas Teuwen

23:31

Doesn't matter, that is just some ugly beast.

BenjaLim

@t.b. Did you think I could not dilate to infinity?

Jonas Teuwen

$$S_N(f)(\pi + \frac\pi{N}) = \sum_{N} a_n \sin(\pi \frac{n}{N}))$$

BenjaLim

yes.

That's what I have

Bryan Dunsmore

Hola mis amigos.

Jonas Teuwen

Some old school messing up, but I shifted the bloody thing.

@BryanDunsmore Sup.

t.b.

23:32

@BenjaLim come again?

Bryan Dunsmore

Nothing much.

t.b.

Hola Bryan

BenjaLim

@t.b. You said I was an algebraist :D

We don't dilate to infinity

user19161

If one cannot dilate, one can dilute.

Jonas Teuwen

Okay, right. Now integral test.

BenjaLim

23:33

DA FUQ I forgot this thing

Jonas Teuwen

Then you have to compute $\int_0^{\pi} \frac{\sin t}{t} \, \text{d}t$.

t.b.

@BenjaLim and a bloody one at that...

Jonas Teuwen

Bleh, just shifting symbols basically.

The interesting question is: why this question?

BenjaLim

@JonasTeuwen I thought I could do it.

t.b.

Homework?

Jonas Teuwen

23:34

Also, does such $c$ exist for other bases? (yes)

BenjaLim

@t.b. Look who posted the question.

I am not a mathematical physicist.

Jonas Teuwen

@t.b. In my course it was more nicely phrased...

t.b.

@BenjaLim I was responding to Jonas's "why this question?"

Jonas Teuwen

The professor -now my advisor- asked me if I succeeded in finishing the exercise. I said: "didn't try, it is boring. Shift symbols" so he was like oh really? And let me do nicer estimates on the Dirichlet kernel.

BenjaLim

Well I solved a problem in Stein and Shakarchi yesterday involving the dirichlet kernel so I though I'd have a shot at this one.

Jonas Teuwen

23:35

I suffered bigtime. And then I loved Fourier.

The $\|D_N\| > \log N$?

BenjaLim

@JonasTeuwen Yes.

t.b.

@JonasTeuwen Oh, that's a wee bit harder :)

Jonas Teuwen

Holy cow.

Without hints?

It took me like a day and lots of skin (nails already goner).

BenjaLim

@JonasTeuwen Well I integrated by parts.

@JonasTeuwen It was not that hard.

Jonas Teuwen

That's not like enough.

BenjaLim

23:37

??

@t.b. what do you mean?

Jonas Teuwen

Really? Then I might have forgotten about it...

In the end you need to talk it to the harmonic series.

BenjaLim

@JonasTeuwen Well yes

but before that you have to estimate $\int_{n\pi}^{(n+1)\pi} |\sin x/x| dx$

Jonas Teuwen

What...? Then how can you not do this.

BenjaLim

that's where I integrate by parts :D

user19161

I don't love Fourier. I love @Jonas.

Jonas Teuwen

23:38

Yes.

@JasperLoy 8-).

@BenjaLim But first you need to do the splitting into this.

BenjaLim

splitting into what?

Jonas Teuwen

You need to write the integral as an infinite sum over annuli.

BenjaLim

no need to.

Jonas Teuwen

Which is like a common trick in harmonic analysis, but if you don't know it it can be quite hard to cook up.

Right. Show me your proof!

BenjaLim

Just note that I can remove the abs value signs

God I have to meet someone at 10

Henry T. Horton

23:39

WHERE'S THE PROOF!?

2

BenjaLim

@JonasTeuwen For $k$ odd we have the integral being

Jonas Teuwen

Start with $D_N =...$

BenjaLim

$\int_{k\pi}^{(k+1)\pi} -\sin x/x dx$

Jonas Teuwen

@BenjaLim Nevermind then.

user19161

@HenryT.Horton I'll give you proof! punches

BenjaLim

23:40

I want to show you how I estimated that

integrate by parts to get that

Jonas Teuwen

Because (for me at least) that estimate is way harder than your Gibbs thingie.

BenjaLim

it is equal to $\frac{1}{(k+1)\pi}+\frac{1}{k\pi}$

plus $\int_{k\pi}^{(k+1)\pi} \cos x/x^2 dx$

So I just need to estimate this integral

Jonas Teuwen

That's not the point, how do you get the integral from $D_N = ...$?

t.b.

Okay, guys. It's time for me to go.

BenjaLim

pffffffffffff

@t.b. bye!!

t.b.

23:42

I'll probably be absent quite a bit in the coming weeks.

See you all

BenjaLim

@JonasTeuwen The dirichlet kernel is $\geq c | numerator of D_N|/|x|$

Jonas Teuwen

@t.b. Good luck with whatever you're up to. Don't forget Scotch, beer etc. Bye!

BenjaLim

You change variables and estimate like a boss to get that $L_N$ is constant times my integra above + O(1)$

I have to go

Jonas Teuwen

Bye.

user19161

I still wonder why most chat users use okay instead of OK.

user19161

23:44

I always use OK myself, because it is shorter and it is the original form of the word.

user19161

Okay comes from OK, not the other way round...

Jonas Teuwen

You've said so before, yes.

user19161

Yes, I like to repeat myself. 8-)

user19161

@henry Are your eyes OK?

Henry T. Horton

Yes

user19161

23:47

You're back @old.

Jonas Teuwen

But the rest of the head...

Rotten from the inside out!

Like mine.

user19161

Now I love to use "My eyes!" everywhere.

Old John

@JasperLoy yep - just moved from one PC to another

Jonas Teuwen

HOLY MONKEY.

Usually pizzas I order are too cold.

Now I have burnt my lip at the first bite!

@JasperLoy "My eyes! Burning in their sockets!"

Old John

time to sleep - g'night all

user19161

23:49

@OldJohn Night.

Jonas Teuwen

Good night.

Henry T. Horton

Moving to that other PC really wore the old guy out

user19161

Yes, unlike Henry who has big biceps.

Henry T. Horton

Embarrassingly large

user19161

Pics or it did not happen.

Henry T. Horton

23:54

user19161

@HenryT.Horton Are you hiding behind them?

Henry T. Horton

I'm in the fanny pack

David Wheeler

dude. let me put you in touch with this guy i know at the soylent corporation....

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$Mathematics$ Mathematics