Mathematics - 2012-03-24 (page 1 of 5)

N3buchadnezzar

00:05

Short q

How do i normalize a function ?

I have V(x) = x^2*(-2*x+(1/4)*a)

But I want V to go from 0 to g, where g is some factor I decide.

So Max(V) = g

Asaf Karagila

00:59

Okay.

I've finished three sections out of the seven planned.

Tomorrow I think I can finish this.

Now I can go to sleep. Goodnight.

Tim

Goodnight! @asaf

Does someone know why "the space $M_1$ of probability measures is included in the dual of the space $C_b$ of bounded continuous functions" in Didier's reply? Any reference on the dual of $C_b$?

I will cook dinner and be back soon.

t.b.

01:43

@robjohn very nice

Rajesh D

Hiya

Hi @tb

still awake ?

t.b.

obviously :)

Rajesh D

I thought you had the habbit of chatting in the sleep...like walking in the sleep :-)

t.b.

yeah, it's like LaTeX, in my vegetative nervous system

Rajesh D

I always wondered...How people on M.SE take so much pain in typing up illustrious answers in good Latex formatting.....I wanted to say hats off to them for their patient efforts.....But little did I knew that they had it in their Vegetative nervous system !

@tb

t.b.

01:54

@Tim It's a version of Riesz representation theorem.

@RajeshD at some point it's rather automatic... :)

Rajesh D

@tb : I got a thought when i was reading Do Cramo book, Is there any way we can compute the topological properties (like number of closed loops which are prominent, etc) of online handwritten characters

By Online handwritten characters we have the X and Y co-ordinates of the pen tip sampled evenly in time

Zhen Lin

Huh, "kernel" isn't French. Another maths word to learn...

t.b.

Noyau

Rajesh D

02:09

French liquor ?

Zhen Lin

No, kernel.

t.b.

For both, kernel in the sense of algebra and in the sense of integral kernel.

Zhen Lin

And, apparently, operating systems too.

or "exploitation systems", hehe.

Rajesh D

Kernel is the central (core) part of a system which is responsible for the behaviour of the system....like kernel of an operating system...Linux has a kernel written in 'C'

t.b.

Oh, if you really want to suffer, read computer stuff in French. Impossible. Octet for byte, for example.

Zhen Lin

02:12

I see...

Rajesh D

@tb : Octet seems more natural than a byte

t.b.

@RajeshD It is not entirely clear to me what you want to do, so you want to enter a character using the mouse as they do here, for example? And you want to determine the number of holes, self-crossings, etc? I'm sure that it shouldn't be too hard to implement.

Zhen Lin

Huh, do they not say co-égalisateur in French?

t.b.

I don't know what they use. I don't think I've ever seen that word in French.

Zhen Lin

Google gives <1000 results. I suspect they just say conoyau.

t.b.

02:16

For cokernel, yes. But for co-equalizer, I don't know.

Zhen Lin

I'm looking at Demazure and Gabriel now, and they say conoyau where I would say coequaliser. But then again, it seems most algebraic geometers do that...

t.b.

This guy likes exclamation marks

Zhen Lin

Yes, yes he does.

t.b.

Oh, did you know that Verdier's thesis is now online?

Zhen Lin

In PostScript? Someone must have gone to a lot of trouble to retypeset all that...

t.b.

02:23

Maltsiniotis did. It appeared in the mid-nineties in the astérisque series.

Zhen Lin

ah

t.b.

Okay, time for me to leave. Good night!

Zhen Lin

Goodnight!

Jonas Teuwen

07:30

Hello Boyos.

Kannappan Sampath

Hello!

Jonas Teuwen

What time is it there?

Kannappan Sampath

1305 hours

Jonas Teuwen

Oh, quite late already.

Kannappan Sampath

Yes, but did not manage to get anything done! : (

Benjamin Lim

07:41

Hey guys

@JonasTeuwen I have an analysis question

I have been thinking about topologies induced from metrics on functions spaces

Jonas Teuwen

Okay, yes.

Benjamin Lim

@JonasTeuwen Say we equip $C[0,1]$ with the sup metric

Jonas Teuwen

Yes.

Benjamin Lim

what is a basis for the topology here? I mean like what do the open balls look like

Or for example $C[0,1]$ with the $d_2$ metric

$d_2(f,g) = \sqrt{\int_0^1 |f-g|^2}$

Jonas Teuwen

Oh, right. Well, for those spaces it is actually better to work with your metric. So your balls are scaled and translated versions of $\{f : \|f\|_{\infty} \leq 1\}$.

Benjamin Lim

07:49

why norm less than 1?

Jonas Teuwen

I just take the $1$-ball.

Benjamin Lim

Oh ok

Hmm this may not form a basis for the topology

Jonas Teuwen

Sure, you need $\{f : \|f - g\| \leqslant \epsilon\}$.

Benjamin Lim

I have a feeling that the topologies induced by the two metric I stated above are not equivalent....

Jonas Teuwen

Yes.

You can make $\int_0^1 |f|$ finite while $\sup |f|$ is infinite.

Oh, on $C[0,1]$, not $C(0,1)$.

Benjamin Lim

07:56

right,

Daniil

hi

Jonas Teuwen

@BenjaminLim You can take for example a Fourier series that converges in $L^2$ norm but not uniformly 8-).

Benjamin Lim

I'm not so advanced jonas

Jonas Teuwen

Okay, so what you need to do is find a sequences of functions that converges in one norm but not in the other.

Benjamin Lim

@JonasTeuwen How can sup $|f|$ be infinite?

Every continuous function attains it's maximum and minimum on a closed and bounded interval

Jonas Teuwen

08:01

@BenjaminLim It cannot be infinite here as you take it on $[0,1]$ and not $(0, 1)$.

Benjamin Lim

@JonasTeuwen Oh sorry your remark was for $C(0,1)$.

Jonas Teuwen

@BenjaminLim $x^{\frac{n}2}$ I think works. Converges in $L^2$ but not uniformly.

Benjamin Lim

I'm trying to think about this in terms of balls containing each other

Jonas Teuwen

You shouldn't.

Matt N.

Morning.

Benjamin Lim

08:08

hmmmm

Matt N.

@JonasTeuwen How do you come up with examples?

Jonas Teuwen

@MattN Hi. What do you mean?

@BenjaminLim This is an analysis question, if you phrase it in topology language it might actually become harder to answer!

Benjamin Lim

hold on

Matt N.

Well if you have two spaces with two different norms and you think they aren't equivalent.

Benjamin Lim

I'm working on something

I am trying to think of an example

Jonas Teuwen

08:15

@MattN Hmm, I was thinking "what sequence does not converge uniformly?"

Matt N.

What does $x^{\frac{n}{2}}$ converge to? It doesn't seem to converge even pointwise so how can it converge uniformly?

Jonas Teuwen

On $[0, 1]$.

Matt N.

Doh.

Jonas Teuwen

It does not converge uniformly.

Matt N.

Thanks.

Jonas Teuwen

08:17

But it does converge in $L^2$.

Matt N.

Well on $[0,1]$ its limit is $0$ everywhere except at $1$ where it's $1$.

Jonas Teuwen

Yes. So uniform fail.

Matt N.

So it does actually converge pointwise.

Jonas Teuwen

But not in the $\sup$ norm.

Asaf Karagila

What about the $\mathrm{soup}$ norm?

Matt N.

08:19

Yes it can't converge uniformly if the limit function is discontinuous.

Benjamin Lim

@JonasTeuwen What about this

Consider the function $g$ that looks on the interval $[0,1]$ like

flat, then a spike high up to two, say and then flat again

Jonas Teuwen

Why take something harder?

Benjamin Lim

For any given $\delta$ I can always find a $g$ such that $\int_0^1 |g|^2 dx <1$

Simply by making the function thinner and thinner

But then if this function has sup of $2$

Then taking $\epsilon = 1$.

Shows that such a $g$ will never be in the ball of radius $1$ about the zero function in the sup metric

@JonasTeuwen Is my last sentence right?

Jonas Teuwen

Why are you so obsessed with those balls.

Benjamin Lim

@JonasTeuwen I am more used to topology

Am I right no?

I think it is correct

Jonas Teuwen

08:22

I'm not sure why it wouldn't be equivalent with your argument.

Equivalent allows it to be "inequal up to a multiplicative constant".

You probably want your spikes to increase in height and keep the same area.

Benjamin Lim

No I think I want the height to be the same and the area to decrease no?

I want to show that there exists $\epsilon > 0$ such that for all $\delta > 0$,

$B_\delta^{L²} (0) \not\subset B_\epsilon^{\text{sup}}(0)$

Jonas Teuwen

You want the $\sup$ norm to blow up and the $L^2$ norm to stay the same for example 8-).

Oh no, the $\epsilon$ $\delta$ stuff.

I only use that when I need to actually prove it.

Not when I think about an example!

Benjamin Lim

So Jonas

Jonas Teuwen

But I can understand that you do that, I did that too 8-).

Okay, can you tell me what you're exactly trying to do with that function?

Benjamin Lim

see, please try and see from my viewpoint

Jonas Teuwen

08:26

Yes, I understand. Sorry :-).

Benjamin Lim

So I want to produce a function in the left that is not in the right

Jonas Teuwen

So a function that is in $C[0, 1]$ with the $L^2$-norm (the left?) and $C[0, 1]$ with the $\sup$-norm?

Benjamin Lim

yeah in the left but not the right

Jonas Teuwen

Okay, good. So what is your function?

Benjamin Lim

the spike thing I told you

Asaf Karagila

08:28

Spike Lee?

Jonas Teuwen

(Suggestion: What does it mean for vector space norms to be equivalent? They have the same convergent sequences)

@BenjaminLim That's not really... clear.

Can you make a small picture of give me a function description?

Benjamin Lim

^

$_ \Lambda _$

$\Lambda$

@JonasTeuwen It looks like that

Jonas Teuwen

Okay. I understand, but from where to where?

Benjamin Lim

what do you mean by where to where?

Jonas Teuwen

Where is the peak, where is it $0$.

Benjamin Lim

08:32

The peak is say at 1/2

The function is flat

then the upside down V thing

and then flat again

Jonas Teuwen

Meh. That doesn't completely determine the function, right?

Anyway. It is in both spaces.

Benjamin Lim

Why is it in both balls?

Asaf Karagila

snickers

Benjamin Lim

You see I want to show that there is an $\epsilon >0$ such that for all $\delta > 0$, I have a function such that it's $L^2$ norm is less than $\delta$ but that it's sup norm is not less than $\epsilon$

Jonas Teuwen

Because the $\sup$-norm is finite and the $L^2$ norm is finite because the function is bounded.

Benjamin Lim

08:35

But Jonas

If say I choose the peak function to be such that in the sup norm it is less than any given delta

Jonas Teuwen

That does not give a counter example, you would want $d(f, 0) \leq C d'(f, 0)$ to not hold.

At least, not uniformly in $C$.

That's what equivalent means to me: They give the same Cauchy sequences.

Matt N.

Ello : ) How's the head?

t.b.

'ello! It is tired but nothing else, as far as I can tell without coffee.

Matt N.

Good. Finally!

Jonas Teuwen

Kopfschmerz? Talk me about it...

t.b.

08:42

I think Ben's example shows what he wants if I understand correctly... He has $\|f_n\|_2 \to 0$ while $\|f_n\|_\infty = 1$, so there can't be $C$ such that $\|f\|_\infty \leq C \|f\|_2$.

Jonas Teuwen

But he has no sequence.

I do know that example, yes...

t.b.

Oh. I haven't read attentively enough. I automatically shrank the base :)

Matt N.

@JonasTeuwen -> tell me about it

t.b.

Everything else would've surprised me.

Jonas Teuwen

Yes.

Plus I don't really understand why there must be balls.

t.b.

08:44

Why is there such a spate of questions on the Hausdorff distance recently?

Speaking of balls: you certainly know that the unit sphere in Hilbert space is contractible, right?

In fact something much stronger holds: it is diffeomorphic to the Hilbert space itself!

Jonas Teuwen

Yes.

t.b.

I sense excitement :)

Jonas Teuwen

Sorry, the unit sphere is diffeomorphic to the Hilbert space?

Hmm, so how would the diffeomorphism look in $L^2$?

t.b.

Oh, it's not very explicit. You start with a kind of stereographic projection and perturb it to take care of the missing point. It's quite non-trivial.

Jonas Teuwen

Oh. :-). I'm not surprised.

Asaf Karagila

08:49

@JonasTeuwen BOO!!!!

Jonas Teuwen

Is this an old result?

t.b.

Bessaga, somewhere in the sixties.

Jonas Teuwen

Ah. Let's see!

t.b.

I guess it's one of those things that you can prove by thinking very hard once you know that it is true, but how on earth should you have the idea?

Jonas Teuwen

8-). I often have that question...

t.b.

08:51

at.yorku.ca/cgi-bin/…

Jonas Teuwen

Okay, the next thing I was wondering: Does it hold for Banach spaces? Apparently: almost.

Hmm, MathSciNet does not give me the PDF 8-).

"Every infinite-dimensional Hilbert space is real-analytically isomorphic with its unit sphere"

t.b.

I'd have linked you to it if it were available online.

Jonas Teuwen

Oh right 8-).

Let's see what the title I quite shows.

t.b.

What do you want to see?

By the way: Maybe! You! Want! To! Go! After! A! Bounty!?

Jonas Teuwen

A bounty! Maybe I do, but I'm still in the process of awakening. Let's see 8-).

@tb How the proof goes.

robjohn

09:10

@tb Mmmm.... bounties...

Rajesh D

Hi @tb : Morning I had too leave abruptly...thats why i couldn't reply you

robjohn

09:24

some annihilation?

Asaf Karagila

Huh... I didn't know that Theo was a Chinese guy, always thought of him as a Swiss (also known as "Zi Gut German").

Matt N.

Bakery time. See you in a bit.

robjohn

@AsafKaragila Chinese?

Asaf Karagila

@robjohn Chinese.

N3buchadnezzar

Heya =)

Kannappan Sampath

09:44

Group Theorists, Can you help me talk about the conjugacy classes of subgroup generated by $\langle (12345),(2345) \rangle$ in $S_5$.

There was question where answers from GAP exist while not from Group theory.

And, Hi all of you!

Asaf Karagila

One of the guys in my office is in algebraic graph theory. What they do often is calculate things in GAP then try to derive the results from the theory.

We often mock him for that. :-P

Kannappan Sampath

Yes, even I don't like doing things with GAP.

Groups are not so computationally taxing, like the finite dimensional projective planes or $t$-designs.

Asaf Karagila

Finite things are often boring.

Kannappan Sampath

Hmm... No, not for me atleast!

t.b.

I started out in geometric group theory. There we consider finite groups as trivial...

At least virtually so.

Dharmendra

09:49

Salem to all mathematician.

Kannappan Sampath

Well, finite groups are interesting in their own right. We don't still know many things about them!

t.b.

I didn't say otherwise :)

Hi Dharmendra

Dharmendra

Hello

Kannappan Sampath

@Dharmendra Hi

Asaf Karagila

I also don't know if the empty set exists or not, which is the most finite thing possible.

Dharmendra

09:51

Good day all

t.b.

@AsafKaragila I think it has 27 elements.

Asaf Karagila

@tb Maybe so.

t.b.

one for each dimension in string theory and then one for me.

Kannappan Sampath

Geometric group theory, I think is the focus of PCMI this summer.

Dharmendra

t.b.

09:53

It's one of the topics that are quite underrepresented here and on MO, less so on MO

Asaf Karagila

@Dharmendra ???

@tb But... geometric things... eww.

Dharmendra

Focus on Step1
I know only angle of the camera
How to find the distance and the height ?
looks like impossible

Kannappan Sampath

@tb It is quite interesting, I heard my supervisor tell me. He says, techniques are quite advanced, asked me to defer attending PCMI, although they wanted only one semester group theory course.

t.b.

@KannappanSampath the group theory itself that is used is pretty simple. But there's a lot of metric geometry and (algebraic) topology being used. Also, the interesting examples come from many different sources, that's probably why he described it this way.

Kannappan Sampath

@tb Quite likely. But I trust my mentor in these aspects. : )

t.b.

09:58

@Dharmendra it is impossible to determine height and the distance from the angle only. but don't you know the height in which the phone is held?

Kannappan Sampath

@tb So, I know the prerequisites, now! Thank You for telling me about it!

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