Mathematics - 2011-10-23 (page 4 of 5)

Right.

7:19 PM

now a continuous linear functional on M(X) = C(X)* with the weak* topology is...

an element of C(X), rather

Yes, I misread.

unwinding all, exhibits the tautology in all its glory...

on the left hand side we have just plain \int f d \mu

where f is a continuous function on our compact metrizable space

and on the right hand side we have mu pushed forward via x -> \delta_x to the extremal points, we have f considered as continuous linear functional evaluated at the point measures, so f(x) and we integrate that over \mu, so both sides are equal

and \mu as a point of K is the barycenter of \mu pushed forward to the measures on the extremal points of K = P(X).

Okay I see that (I'm sorry, I don't really dare to give full answers if I'm not totally sure and the "opponent" is much better).

The fun of it is that this tautology is actually useful!

How?

7:27 PM

It is implied by everything!!!

Take a measure preserving continuous transformation of a compact metrizable space

(the given measure is a probability measure)

Okay, then what? (I had assumed that it was a probability measure since that were the only measures which I have seen in the context of measure preserving transformations)

Okay: Now the set of all probability measures is a compact convex subset of all probability measures

Its extremal points turn out to be exactly the ergodic measures (think about that later).

Cute.

Now Choquet gives you a measure on the set of ergodic measures

that represents your original measure as an integral over the ergodic measures.

That is, you have a means of decomposing your measure into its ergodic parts

7:33 PM

Oh, that is a nice result.

It's called ergodic decomposition

Ergodic theory? Oy vey.

And you can often use that to reduce questions about general measure preserving systems to the consideration of ergodic systems

Okay, that is nice. I don't think I saw that in the book of Einsiedler and Ward... (Or I have skipped it)

So far this is pretty much too soft to be really useful, but you can push the argument further to actually decompose your compact metrizable space into components on which your transformation acts ergodically. The measures on those components are those given by the measures arising from Choquet's theorem.

That's pretty much the showcase application of Choquet

(at least to my mind)

7:39 PM

@AsafKaragila I was tempted to make a joke, but it would be against the rules ;-)

@robjohn What rules? I only recall setting one rule: Set theory or non-math discussions!

@Asaf: There's a lot of non-trivial descriptive set theory going on in these ideas. Kechris did a lot of work on those things over the last twenty years

@tb We are only at Chapter 3 as yet ;-).

I know. I'll stop now :)

Figures as much. If it's deep, there's set theory involved ;-)

7:42 PM

*hands over the scepter to the AoC crowd and its König*

lol

Speaking of descriptive set theory.

A friend of mine whose research is C* algebras is going to a descriptive set theory conference. I find it completely unfair.

Why don't you join him/her?

I have no funding, the requests for support is closed, and I doubt that in three months I can get a US visa, while he's got an American nationality as well an Israeli one.

You can't get a US visa within three months?

I think it takes up to two months. I don't know.

7:48 PM

So? 3-2 = 1 > 0

It's a shitload of trouble to get a US visa for people that need a visa.

That's another issue, then :)

Oh, sure. 3-2=1. However by the time I get to actually do that... -1-2 = -3<<0

@AsafKaragila recently I've got it in 3-4 days

and the service was perfect

The funding issue seems more serious

7:49 PM

Yeah, much more. Especially since a ticket to the USA is a small fortune.

icic to quote mr G.

About 500€?

@tb that's me?

Had it been a ticket to Europe I might have managed with some money from my dad at the worst case scenario. USA? No way.

Doesn't your professor have some money left?

7:50 PM

who else? Gobjohn?

;)

Well, he got 0 funds to begin with...

Okay, that sounds like a real obstacle, then

@tb who is it???

@AsafKaragila pity to know

Well, that sucks, Asaf.

@Gortaur I only saw you use "icic" in here. nothing negative implied here

7:52 PM

@AsafKaragila I seem to remember somewhere a mention about inappropriate topics, but I can't find it anymore.

Yeah. Which is why I really want to finish this proof with that guy from TAU this week. Having an actual theorem would give me the opportunity to present in a conference and then get more funding.

@tb and who is timid now? ) I can live with it to quote mr. T

I pity the foo'

@Asaf ?

The A-team!

7:54 PM

Mr. T!

@tb I was just curious who is Gobjohn. And by the way, I learned icic from my friend from Singapore

I find it extermely useful

@Gortaur I unmasked myself; are you going to stay hidden?

@Gortaur Oh no! We are becoming one!

@robjohn oh, that was the reason? sorry, I think I said nothing about this

Gobjohn!

that more about you than about me :/

7:55 PM

@robjohn: Hahaha! Love your profile page!

my mind to your mind, my thoughts to your thoughts

is it a wedding oath?

@AsafKaragila Asaf ?

There are two questions with 4 closing votes...

@Gortaur Gor?

@Gortaur Have you watched the original Star Trek?

@AsafKaragila Where can you see those?

7:57 PM

@AsafKaragila Gor*t*

@robjohn nope, that was not popular in Russia in 90s

It's what Spock says when performing the Vulcan Mind Meld®

@JonasTeuwen Moderator Tools.

then I didn't have a chance to watch it

Oh, I still need 6025 rep to get those...

Patience, young grasshopper.

7:58 PM

@JonasTeuwen 4k guys from Delft )

Yep.

@tb what have you guys done to my compatriot?

I will not admit silence after such question ]:->

@Jonas: this one and that one I've already voted to close one of them

@Gortaur he just disappeared.

math.stackexchange.com/questions/67913/types-of-markov-chains that one is still alive?

@tb Some people never cease to amaze me here.

8:04 PM

@tb that's the usual justification for such incidents, isn't it? )

@robjohn Moi?

@tb Closed one, voted already on the other.

@AsafKaragila No, check the link to the comment. It is from a while ago.

@AsafKaragila unless you were one of those who upvoted that question.

What question?

the question that was linked to in my comment to t.b.

Ohhh that. I did not upvote, only LaTeXified.

I also voted the comments too. :P

8:08 PM

The people who upvoted it amaze me.

Sorry, I'm lost

God, I am being soooo cynical with that "inherently closed question" post.

@tb I commented on something you said a while back and it was confusing Gortaur. You have to click on the right arrow to see what I was referring to when I said "Some people never cease to amaze me here."

Yes, I don't understand that. Really. There were too many parallel threads for me.

Inability to do multitasking. I guess that proves my internal XY chromosome disposition...

On the line where I said "Some people never cease to amaze me here." at the left of`@tb` there is an arrow, yes?

8:12 PM

I figured it out by now, thanks!

I was already commenting on your comment

And I was trying to unwind all the links so that Gortaur didn't think I meant him.

math.stackexchange.com/questions/75091/your-minds-are-too He changed the title of the question.

That is borderline defacing the question, in my opinion.

<_<

>_>

:D

@Asaf: Borderline?

Way to go Arturo!

Yes. I was considering to vote to re-open because the question isn't that bad but the behavior of the OP made me reconsider

8:18 PM

I would consider to delete it altogether.

However it seems that the community not yet trust me in full, so I have to wait like two whole days.

Like with that other question, that I could finally vote to delete...

I don't think it's a good idea to be overly quick with such actions

@tb No premature eradication?

Usually I would agree. However in this case the writer seems to have some very bad etiquette to begin with.

And well, I am a big fan of iron fists. And drunken boxing too ;-)

Thing is: if you let him perform his trollish actions on this question you have it localized. Maybe he even considers removing some of the comments, etc. If you remove this playground he might post a new question, it'll take time to close that one, etc.

Is it just me or have we seen a lot of "tail of the normal distribution" questions lately?

8:35 PM

I thought eternal September was over, but there are only the occasional gems now. The site is a lot more active, but the average questions are distinctly worse than before. Probably has to do with the beginning of the academic year.

Just my 2 cents

@tb I hope you're right. That way there is hope that it might return to a better average question.

e.g. I just answered a question about why the square of the distance is not a metric.

There are too little analysis questions.

@Jonas: here's one you might like

I am tempted to retag as descriptive set theory :-)

What bothers me is that people don't show thoughts at the moment, just as the one I just linked to. part 1 of the question is trivial, the key to part 2 is to show that Baire class 1 is the same as \Sigma_2^0-measurable.

8:49 PM

what is "pointwise discontinuous"?

I don't know. wrong summary of what OP actually wants to show

@tb That is a weird question... But what is \Sigma_2^0?

(weird as in: most books on measure theory should have this)

functions whose points of continuity form a dense set.

I have never heard that before.

That is \Sigma_2^0?

@Jonas: I wrote that for Asaf: \Sigma_2^0 = F_\sigma-sets (countable unions of closed sets). See Borel Hierarchy

8:54 PM

No, pointwise discontinuous

Oh, F_sigma.

I got that from A Historical Gem from Vito Volterra

It seems to be used a lot around the turn of the 20th century.

Why is the F_sigma (Fermé, somme) a French abbreviation and G_delta (Gebiet, durchschnitt) a German one?

I think it was a compromise a the end of WWII :-)

@robjohn: Off a few decades :)

8:58 PM

Before WWII?

@t.b.: to get a laugh, you have to break a few legs.

Break a few legs...?

@JonasTeuwen There is a saying that to make a soufflé, you have to break a few eggs...

Oh, like that. Thanks.

@Jonas: It goes back to Hausdorff's Grundzüge der Mengenlehre 1914. He used that notation, but why, I don't know.

9:01 PM

some say omelette, but you get the idea.

and also for a performer, they say "break a leg" for good luck. So putting them together...

Thanks, @tb.

It is kind of nice to use consecutive letters for related things. That's why we have H(M) (homology) and K(M) for K-theory (I and J were deemed unsuitable by Grothendieck)

And G would have seemed presumptuous, I presume

Why were I and J unsuitable? :D.

@JonasTeuwen Which girlfriend, the one who didn't flinch at your Russell lecture?

Howdy J.M.

9:07 PM

(Good grief, I fell asleep at the keyboard... )

Heh. Here is an example of the questioner not understanding the answer he accepts.

@JM you have the right to fall asleep at the keyboard after creating that fabulous gravatar!

@robjohn: unbelievable...

@JM Yes.

Discrete math? Really?

@rob: Here is another.

9:09 PM

But the pointwise discontinuous guy invested some effort

Hey rob, nice shiny new badge you have there... :)

He gets my vote of confidence for this effort!

@robjohn Funny.

@JM I knew your appearance was impending. There were starred messages popping up out of nowhere. :D

@JM I may sign up on WordPress and set it as my gravatar.

9:12 PM

Jeez, I remember a time when 25 messages was a lot to wake up to... :D

@rob: Oh, the Mean Square. Go for it, man!

@JM Oh, headslap you meant the Outspoken badge (duh!)

"math.stackexchange.com is not a black-box." - Maybe we should say this a lot more...

@robjohn Nice how you get the meaning, Mr. Mean... :-)

Could I query it on data.SE to find out how for off I am from the Outspoken badge?

At least now this place is starting to sound like the faculty lounge I'm used to... :)

9:16 PM

@AsafKaragila It may take me a while, but I can usually hit my head against the wall enough to get the gist.

;-)

Hooray! I got more reputation than Arturo!

(On MathOverflow, at least)

Very good. Can I touch you?

That pointwise discontinuous function question will be a major headache to answer for whoever wants to do that... It seems to me that the entire argument is essentially there, so you have to follow that route and fill in some minor technicalities.

Only if you're my girlfriend. Or if you slip me a Mickey Finn in my drinks when I'm not looking ;-)

Mickey Finn is passé; roofies are the rage nowadays... ;)

9:26 PM

@Jonas: and then there's this one which you may want to answer. I don't understand the hurry that guy is in. I would never have understood Hille-Yosida and the ideas there in a single day...

This is rather peculiar; one would think somebody would have been taught this limit way before he hears about series expansions...

At least he picked up Engel & Nagel :)).

I remember correctly and you can't prove sinx/x at 0 using L'Hopital, right?

why not? of course you can

but you shouldn't

sparrows and cannons and the like

I remember it was somehow used somewhere or something in the proof, and using L'Hopital would cause some circularity or something like that...

9:30 PM

Asaf likes that.

I remember very very vaguely.

@Jonas: yes, didn't seem to appreciate my suggestion, though

Hmm, depending on how you define sin/cos that might give circularities? (for example when computing D sin x).

precisely.

@tb Too bad.

9:32 PM

@Asaf: the only thing you shouldn't do is to use it for monomials

that enters in all the proofs I know

I see.

Oh yeah, there was a question about x^n \exp(-x) asking if l'Hôpital could be used...

a few hours ago

Advice, please: Am I giving this guy too hard a time if I post (after bugging him twice about a reference): "Thanks a lot! Yes, I did really want to know. I also think that if you reproduce a major passage from somewhere you should definitely say where it's from."

Ooh, I capped and got a nice answer badge just now.

Congratulations!

9:37 PM

Ditto.

@t.b. Why would he be so coy about that book? It's a nice one...

@tb Doesn't sound too hard.

Okay, thanks, posted (with a minor modification)

@Asaf: back to sin x / x: usually you define the sine as a series then factor out an x in the numerator and cancel, there you are.

haha my gravatar came through, too :-)

@robjohn Lovely.

The little sized one looks like an alien looking through a small square window.

9:47 PM

Did you draw it yourself?

scares the bejesus out of me

@JonasTeuwen Yep, this morning in response to a comment by t.b.

Cute.

10:02 PM

Do we really need the (greens-functions) tag?

@JM woow three new tags for one post. That must be a record...

In answer to your query, I don't think so

But he is a good user, wouldn't he feel "insulted" if we remove them?

Whew. Is Mike a troll?

@Jonas: He shouldn't be...

@rob: after checking his activity... hard to tell.

Some people are even worse...

10:09 PM

(FWIW: what you did is more or less what I remember was done way before we started differentiating trigonometric functions...)

@JM He is certainly picky about definitions. I checked to see if he also had an answer, but he doesn't.

@JM Yeah, and I've posted that proof at least a couple of times on sci.math.

@robjohn: I'll remember to put a bomb in the mouth of the horse I am going to give you as a gift when you reach 10K ;-)

I am not bent out of shape about it, I just don't know what to say to him other than what I did.

@robjohn: I don't think so, honestly. looking at his activity he's usually pretty polite, also recently. Wait and see what he responds

@Asaf: there is a wonderful Firesign Theater routine where you hear someone exclaim in the background, "will you look at the mouth on that gift-horse."

10:12 PM

math.stackexchange.com/questions/75207/euclidean-distance-proof What does this have to do with discrete mathematics?

@Jonas: I know, right? Do the honors?

@Jonas: nothing, probably from a discrete maths course

Okay.

I didn't plan it when I chose the color, but when on a black (or dark grey) background, such as when you hover over a gravatar below an answer, it is in keeping with the upcoming Halloween holiday colors. :-)

Oh, Halloween, my birthday (almost)

10:18 PM

@tb one of my friend's daughter's birthdays is Oct 30

Hmm, it is nearly Halloween... my, how time flies.

@robjohn: same here

They started to import Halloween here, and now you can't go out if you don't like spiders and spider webs and all the other scary stuff...

Cool! She'll be thrilled -_-

speaking of trolls. Did you see the two closed meta threads on meta.MO? Neutrino questions hah! here you become tumbleweed :)

A rather rough start.

10:25 PM

@tb they didn't celebrate Halloween there until recently?

Q: A question about "parameterized" sets

Hooray! I proved P=NP!! :-)

-1

Given two sets $A$ and $B$, I want to know the result of: $$C = A \setminus B$$ If I define the sets $A(n)$ as the set such that: $$\begin{align*} A(n) \setminus B &= \emptyset; &\quad &\forall n= 0,1,2,3,\ldots,\infty &\quad &(1)\\\ \lim_{n\to\infty} A(n) &= A &am...

elementary-set-theory

speaking of rough patches, Spoonwood's back

@robjohn: nope. We knew about the concept in elementary school, certainly, but we did nothing special on those days.

@anon: No worries, methinks the mods have tabs on him.

@AsafKaragila Let me know when you get your Field's Medal. I will come to see you accept. :-)

2

10:28 PM

@J.M. I think he has some social communication issues but I was a bit unsettled about him getting suspended.

I am inclined to think that the OP has a problem in the definition, @rob.

@anon: I wasn't terribly sure about the suspension myself. I just think he's being rigorous for rigor's sake, not because he actually wants to do *something*.

@AsafKaragila Really? And I was getting packed for Seoul. :-(

Don't worry. I have like 14 years to make that huge impact on mathematics ;-)

@AsafKaragila That boat sailed for me 12 years ago.

10:33 PM

damn, Didier is quick

Yes, he doesn't waste any time chatting.

Or packing for Seoul...

Well, he beat me by a few seconds, then I decided to remove my answer because it gave more away than his.

Anyone here is fluent in computability or complexity?

ah, the mean square

as they say on irc, "Don't ask to ask: just ask."

10:36 PM

@tb The early bird gets the worm/.{bird->answer,worm->upvote}

@anon: I just want someone to help me find the problem with the guy who claims I proved P=NP...

@robjohn Allow me to find a fortune entry you just reminded me. One that I like a lot.

@robjohn: I tend not to upvote such answers, there are enough people who do.

@Asaf: Where?

Every morning in Africa, a gazelle wakes up. It knows it must run faster than the fastest lion or it will be killed. Every morning a lion wakes up. It knows it must outrun the slowest gazelle or it will starve to death. It doesn't matter whether you are a lion or a gazelle: when the sun comes up, you'd better be running.

The best fortune my wife and I got was "If you satisfy your thirst, you will soon to be received."

10:38 PM

I don't see why you have proved P = NP but anyway: congratulations.

@anon: The question I had linked just a few posts ago on the chat.

oh

@anon what about the mean square? He's better than average.

You really expect that, don't you?

@robjohn I'm sorry but I don't understand what that is saying. Googling for it leads me to Google gulp

10:41 PM

I think he's a kind of a square, personally.

@tb It means nothing. It is so devoid of meaning that we love it :-)

Ah, Google Gulp. Gotta love poisson d'Avril...

I wonder if he really did prove that P=NP in that paper...

On another tack: I see that the bottommost question on the front page was changed 4 hours ago. I'm not sure I'd be excited about the day when we reach the same level of activity as SO...

Poisson d'Avril

10:45 PM

@rob: precisely that poisson... :)

@robjohn: thanks for the clarification. I know my English isn't perfect, but I was beginning to have doubts... pfuuh

I never knew that was the name for April Fool's Day

Nowadays, I just sit back and enjoy reading the Onion on that day; I think I've filled my practical joke quota for a lifetime.

My cat is depressed. I got her a scratcher which has catnip inside, but she doesn't care about it.

So didn't eat much today, nor drank too much. Her fur feels weird as well...

...how many days has she been that way?

10:52 PM

Three-four days, with the food intake varying but not too much either way.

If she's still that way the next day... time for the vet.

(If she's ignoring catnip, something seems dreadfully wrong.)

I need to check how this scratcher works. It has a bag of dried plants (supposedly catnip) in the pack, but it does not say how to apply it. In fact it says that the catnip is already in the cardboard scratching board.

And that the owner needs only to scratch it a bit on its own to release some of the catnip to attract the cat.

Well, you'll definitely know if the catnip smell is coming out...

So if it smells like a cardboard then I wasted 40 shekels for nothing?

@JM It's getting too much for me now already. I'll have to start filtering out the tags I'm really not interested in. There were roughly 180-190 questions active in the past 24 hours.

10:58 PM

@AsafKaragila Well, you should try scratching yourself first if that's what the manual says...

@JM I did more than a few times. The dried plant had no smell either, and I tried putting it under the scratching board.