@Ilya yes

cool, I have a question related to yesterday's discussion if I can

This guy evidently doesn't want to have an answer...

@Ilya Sure, what's up?

I am a bit confused with kernels - and formulation of the problem can take a couple of minutes

Sure, I'll wait (but it's been a loooooong time that I haven't thought about kernels)

but the problem shouldn't be too hard, just I'm puzzled

So, let $(E,\mathcal E)$ be a measurable space and $K(x,A)$ be a stochastic kernel there. For any $n\geq 0$ on the product space $\Omega_n = E^{n+1}$ with a product $\sigma$-algebra $\mathcal F_n$ we define the family of measures parametrized by $x\in E$ in the following way

render

Sorry, Chinese Internet connection is not good

I'm following so far.

ok, in the latter formula $A\in \mathcal E$ and $B\in \mathcal F_n$. I wonder how to obtain a formula for $P_x^{n+1}(C)$ - which is analogous to the integration on slices we discussed. The indicator in front of the integral puzzles me and I cannot get a clue

@tb: that's my question to you - could you help?

Thunderstorm! Ha, maybe that's the reason for the bad Internet connection :)

Hell, I even don't know if you received my last ping :(

I received it.

my query was not correct - of course you will always think that you've received my last ping :)

Well, in place of $A$ you should have a horizontal slice and in place of $B$ you should have vertical slices, no?

yes, but I don't know how to write it formally - the image is in front of my eyes, but I have no idea how to write it via the integral

no, it's not right because even for a rectangle $A\times B$ we will integrate it over $A$, not over $E$

But won't the integral be zero on $E \smallsetminus A$ anyway?

If $x \notin A$ then $C_x = \emptyset$.

why should $P^n_y(C_x)$ be zero for $y\notin A$?

for $C = A \times B$.

ah, that's because $C_x = B$ only if $x\in A$ and empty otherwise? oh, you wrote it already

@tb: still puzzled a bit - but do you mean that the formula is correct?

Yes, to my intuition it looks correct. If $C$ is an arbitrary set, then $P_{y}^n(C_x) \gt 0$ is only possible if $C_x \neq \emptyset$.

In other words, your characteristic function in the first formula is superfluous, it won't do anything to the integral.

@tb: wait. In my very first formula I have an indicator with $A_0$ and first integral with $A_1$ - now I have both indicator and an integral with the same set. Somewhere should be a mistake

rain! finally...

it means that the first term is in A and the shifted trajectory is in B. Yes, it should be true

I think you want to replace $B$ by $B_x$ and set $B_x = B$ if $x \in A$ and $\emptyset$ otherwise. Then you can drop the $1_A(x)$.

and then I can try to use the same technique with pi-lambda to extend it

exactly.

hey guys

ok, so that was the confusion - I was integrating $P^n_y$ w.r.t. $A$ instead of $E$

@Ben: здароффки

@tb: thank you :)

hey

I'm trying to check if some maps are well - defined

I realised that in the checking I needed to use linearity. I don't think it makes sense to first check linearity then check well definiteness.

@Ilya so, we're all set?

I think this works.

@tb: it should - I will check it now.

stupid conference, I went to take a nap before the dinner which was at 6.30 and went to dinner in time

Basically, it should give what is sometimes called the convolution of Markov operators.

you mean $KL = \int\limits_E L(y,A)K(x,dy)$?

ya, something like that.

Okay, I should go now. See you guys later!

@Ilya I feel your pain... :)

executioners needed

Hi and bye, JM

... :D

@JM I was kicked off the chat after that sentence of yours

@Ilya Tsk, tsk...

executioners also needed

Some trick needed with $\int_{\pi/2}^y \frac{\sin(x)}{x} dx$, ideas?

I am sure I have done this earlier...change to exponent function or was it s/thing else?

Moved the q here.

Here's some cat-egory theory:

Head iz warm if you rests it on power block (in theory).

In practice also.

(The caption is lame but the picture is cute. You can add your own caption if you can come up with a better one : ))

...we're going to do ICHC here? :)

Sorry, couldn't resist : )

But I thought if it's cat-egory theory it's on topic...

*purrs*

I can hear it!

bbl : )

@Matt: hi

@Ilya Nihao : ) See you later!

Nihao-nihao...

Where does this spate of idiotic questions come from?

dunno

unbelievable...

isn't this again the same issue as before?

I don't think that these definition are equivalent

first can be non-zero for $x\notin A$, second cannot

Just slice $C = A \times B$ and replace $P(y,B)$ by $P(y,C_x)$.

Sorry, what do you want to tell me? I need to integrate by slices the second case and I now how to do it thanks to you - I'm just confused that for the definition of probability measures I cannot use the natural way to convolute kernel and a family of measures

@tb: when I finish the draft, could I send it to you - maybe there would be some obvious mistakes in proofs. Only if you have time, of course - there are about 3 pages A4

I don't really feel competent doing this (and I have very little time these days). I can't promise to look at it thoroughly.

ok, sorry if I asked too much

@tb I know and I respect it in people

Thanks a lot.

@Ilya But back to your problem: I think the $A$ at the bottom of the integral is a bit misguided: you let $y$ vary and $A$ has nothing to do with $y$!

@tb wiat :)

if $K$ and $P$ were kernels then we cannot use $A\times B$ as an argument of a product kernel, since its argument has to be a measurable subset of the initial space, not of a product space

So the first formula (with $A$ at the bottom of the integral) I even don't if it makes sense

I'm now utterly confused...

Main question: does the second formula indeed introduce finite-dimensional measures for Markov process?

@tb: sorry for leading you to a confusion

on the other hand, if $P$ does not depend on $y$ and $K$ on $x$ then the first formula define the product measure, while the second does not

@Ilya Hey, I'm sorry, I can't really focus right now. I would try to get rid of the characteristic function as we did earlier today. What happens if in the second formula you write $C_x$ instead of $B$?

(and I should leave in a few minutes)

please, leave if you should - we can talk about this later :) I'm gonna to sleep then, it's 11.30 pm here

@Ilya okay, let's try tomorrow or the day after. Good night!

@tb thank you, have a nice day

(and I hope you sleep a bit longer than last night)

Thanks :) how do I hope...

okay then, see you soon!

@TheChaz Hello Chaz, may I ask you something?

Hi guys.

Yo whatz up?

@Ilya Did you ever see a PhD defense in Leiden? They have this old room full of paintings of long dead professors and there isn't even a blackboard!

@DavidWallace Well, about ... 10 000 EUR.

This is a most intriguing username...

@JM ever seen teeth with IQ?

Hello.

@tb No, I have not. :)

@JM : D

A statement from Lam's Frist Course in Noncommutative Rings. For a field $k$ we have that these two conditions are equivalent: (1) $(a,b,c,d)\neq (0,0,0,0)\implies a^2+b^2+c^2+d^2=0$ and (2) $-1$ is not a sum of two squares. I can see it with "two squares" replaced with "three squares" but two squares don't seem to work for me. Is it a mistake?

@JM He might invent Stupident's t-distribution...

Or tea-distribution?

Hello Matt.

And everyone :)

Hey ymar : )

Hi ymar, hi Matt and hi, J.M.

@MattN Maybe, maybe...

hello and hello, and finally, hello. (pick which "hello" you want...)

@JM Now I need to start balls puns. One would think that's easier to come up with than teapot puns.

@JM Can I have all of them?

@MattN Well, it takes balls to come up with clever ones...

@JM : )

@MattN Don't be greedy. :)

This is quite an achievement...

Oh, it should have been $a^2+b^2+c^2+d^2\neq 0$ above...

I was sitting at a table drinking some beer, but there were only full professors sitting there. So I was like. I'm not qualified to sit here! Then I went to drink alone. Much better 8-).

@tb Why an achievement?

@JonasTeuwen just don't think that six beers from now your feeling that now I'm qualified will be reliable :P

@JonasTeuwen You don't have peers to drink beer with?

:P.

@JM Hmm. I don't know them well enough, and then I'd have to talk.

For some reason I know the professors better...

Oh, I saw Koornwinder last week, he was minding his own business together with a beer.

Well, if all of you have your lips busy with the bottles, there wouldn't be much time to talk I suppose... :)

@JonasTeuwen He's around sixty now, right?

Hmm...

@JM He's emeritus.

I think he's like 67 then?

I'll see you in a bit. Have to go scavenge. Haz teh hunger and fridge is empty. Hope to see you all later.

@Matt:

This is hilarious: the order got messed up now we got 6 toothpastes instead of 1 (like I ordered).

I guess we'll be having toothpaste for dinner... : )

probably tastier than the soup yesterday :)

Eww.. : ) I had to thoroughly scrub my face today to get rid of the smell.

@tb I shudder to think what kind of soup has a taste outdone by toothpaste...

@tb Btw, this reminds me: how many times during your long years of practice did it happen that the towel slipped while you were closing a window and a neighbour was looking? : )

(I'll be back later, now fo' real)

@MattN never. That's what dressing gowns were invented for :)

: )

@JM read here. But free sample powder soup, gee. :)

encrypted-tbn3.google.com/…

encrypted-tbn0.google.com/…

@tb Once upon a time, I would have said "you get what you pay for", but I've managed to be surprised by free samples... :)

I remember back when I was a student that free samples at the supermarket near the university took care of my weekend's lunch.

They weren't bad.

I'm always in for free food.

I always thought there's no free lunch...

@tb Well, really tasty lunch, yes. :) If you're alright with plain palatability, you sometimes get lucky...

If it's edible, I'll take it.

vimeo.com/8746351

Can we take snapshots of pages in DJVu?

@KannappanSampath Depends on your viewer.

@JM I seem to be using DjView @JM

Bleh.

Too much to do.

Drank 500g worth of coffee today. Still sleepy.

@KannappanSampath You should have an "Export As" option in the "File" menu. Pick your favorite image format.

@JM Yes!!!! I can export it also as pdf--amazing. (I never knew this and relied on other online programs for converting to pdf.)

Exporting to PDF didn't want to work very well for me when I had to do it. The text would become unreadable.

Oh, I see.

@ymar It depends greatly on the DjVu also... :)

Less reason to be happy, but thanks for telling me. @ymar

@JM I would guess so, I tried only once.

@KannappanSampath That's just what happened to me. I haven't done any serious testing.

It's admittedly hit-and-miss, but you should gamble anyway. At most you'll only lose a few minutes...

(also some tweaking of the export options for PDF might be necessary)

@KannappanSampath When you have the time, could you please have a look at this and a correction here?

@ymar Will do! Just typing up a book reco for Analysis problems. I like the book very much I want to give a sneak preview.

What's the book?

I have typed an incomplete answer you may want to see here

I should be going. See you people in a few...

@JM see you!

@KannappanSampath Thanks.

Bye J.M.

Bye JM

Bye.

@KannappanSampath what did you want to have? a screen shot?

@tb in the incomplete parts?

I want to add some screen shots of problems from each book. Three pages--one for each volume.

I am seeing dead Jim--that's why I posted that before. I am afraid I'll lose what I type.

I had no idea those books were translated to English. Good call, English speakers!

Yes, the authors are polish.

If it hadn't been for these books, I'm quite sure I wouldn't have been able to go through my analysis courses.

So, one more person feels the same way as I had felt. :)

@tb Did I answer your question?

@KannappanSampath you did. I was on the phone. I'll see if I can do something.

(gimme a few minutes)

Sure, Thank you. I am trying my luck too.

Here's what I had managed--Is that OK? @tb

Does the English version have solutions to the problems?

(I converted to .pdf and chose random problems and put the here.)

@ymar Yes, it does.

Hmm, missed that^

Hello

I just said what you did, but you were quicker. I hadn't noticed that you said in that answer that there were solutions in the book.

Hi, Rajesh.

@KannappanSampath here's another page

(from Volume III)

@tb Thank you. Saved my work. Shall I add this as page for Vol. III into the answer?

@KannappanSampath if you like it, take it. It's already on the stack.imgur server anyway.

@tb I think it's fine. For continuity I would like to choose random problems from both continuity and diff--so a page won't be a good Idea.

But, in Vol III, I have no idea about Leb. Integral, so, this is what I'd have done.

Which books are you guys tokkin' about? Vol III sounds like the Princeton Lectures in Analysis...

Kaczor-Nowak analysis problem books.

No, @Jonas. We were talking about Problems in Math Analysis, (3 vol. book) pub by AMS.

captime!

number one this year...

Congratulations!

Thanks, ymar

@tb Nice. Congratulations.

@tb Fun.

That means you've done too much.

@JonasTeuwen no, today I just answered a very low hanging fruit, and this nice question which allowed me to give the kind of answer I like to give :)

@tb Yay : ) Congratulations!

@MattN Thanks, Matt! So, how did the scavenging go?

Very well. Ordered pizza : )

And got toothpaste? Interesting pizza delivery service :P

Haha, no the delivery was a different one (for food and other stuff) but when I'm not in the mood to do it there is no planning involved and the outcome is that we have stuff that we needed but possibly nothing to make dinner with. : )

And to be fair: I'm pretty sure it's a bug on their website and not the people who pack the stuff into boxes.

@tb You always give very complete answers. Where do you get the stamina/energy? 8-).

Coffee maybe? : )

His coffee consumption is high.

That can never beat mine.

Maybe he loves what he's doing.

I wouldn't be too sure. I've never heard of anyone drink through an entire packet of coffee in two days.

Finally after so much of pain and with t.b.'s help...

It should say "courtesy", you're missing an r there : )

@JonasTeuwen Well, thanks :) I tend to only answer questions I'm interested in and one thing I like doing is to expose results I like, so there's not much energy involved. Another thing I love doing is digging up papers by the old masters and reading through them. I knew the answer to the question more or less right off, but I had to think a bit where the result actually came from. Distilling the ideas helps me keep sharp, I think.

@KannappanSampath Hey Kannappan, I'm sorry for the quaternion balls-up. It was me being lazy who caused the embarrassing situation for you and I'll make sure to be less lazy in the future. M'kay?

Okay, that's a good thing.

Mathemagician really likes capital letters it seems.

@MattN Please do not put me into this. This is uncomfortable and not the quaternions event at all. After all, everybody makes mistakes. So, why worry this much about it?

@KannappanSampath Heh, then forget about it : )

I am OK with it. It's not an issue we need to worry about. But, we need to make sure we are not commiting this mistake again.

On second thought, will AMS sue me for the sneak preview?

I guess you'll find out soon enough : )

I am afraid of this issue now!?!

: D

Don't worry. First they would tell you to take it down.

It makes no sense to sue you from their point of view.

Hah! Nice people. As long as they won't take me to one of those courts and make fight extradition to my country and all my career go down the drain, I am happy.

lol

I think he means the TPB=the parabola, no?

pirate texting on the internet: rrrrrr

@Matt Thank you for the edit. :)

Welcome : )

Automatic analytical continuation, that's SCARY.

Ha, Polish Wikipedia seems to be the only one with an article on Rees matrices. I'm so proud of myself. I've done something nobody in the English-speaking world cared to do!

So you're proud that there is some crazy Polish guy that does something nobody apparently cares about? 8-).

@JasperLoy Hi

@JonasTeuwen I think my exams and thesis are scary.

@MattN Good!

@JonasTeuwen I'm proud because I'm this crazy Polish guy!

@JasperLoy Thanks ;-)

I have no idea how to contribute to Wikipedia.

@KannappanSampath Click on the edit button. : )

For instance, I'd like to add an article on Cameron's Theorem which does not exist at all.

@JasperLoy No, I've edited several instances of abysmal writing.

@ymar 8-).

This is a beautiful answer. :)

@ymar What field are you in?

I have to edit once a year or I'll lose my admin privileges. I got them years ago, but I have to keep them in order to boost my ego by looking at them.

Wikipedia also have a chat room for editing help.

@JasperLoy Yes.

@tb Impressive.

@JonasTeuwen If I'm in any field, I'm in algebra. But I'm busy myself thinking of quitting maths.

@ymar Great. What will you do then?

@JonasTeuwen Sell burgers. I love this idea but my parents will kill me.

@JasperLoy Because I suck at it.

@ymar Sounds like a good idea. I wouldn't care too much what other people think.

@ymar Will you please stop thjinking about it? Are you kidding? Please think when you're not intoxicated. I am afraid this will come across as rude. You do quite an impressive algebra. So, please please...

Selling burgers can be more useful than algebra.

If those are good burgers...

@JonasTeuwen : D

@JonasTeuwen I couldn't agree more.

@JasperLoy The techniques, algorithms and blah....?

@ymar That's not the impression I have from your questions...

Well, not my impression either, but that can only make me conclude that I suck at it as well.

@tb But have you seen my answers?

@tb Me too. Not just that. I have myself learnt a lot from @ymar.

@ymar Yes, may be not on the site; but on chat, yes.

Don't try to convince the guy to not make good burgers.

Thanks guys, but I know how much I know. And this discussion embarrasses me.

@JasperLoy I agree it is good. But, the model of the text book I'd say is adapted from Van Lint and co.

This conversation makes me hungry.

Missed the punchline, but I don't think I know the guy.

Okay, you're probably not really into operator theory?

Not at all.

Okay.

@JasperLoy I'm not sure if that's not better. :)

He does stuff like "topos theory and quantum logic". I haven't have the slightest clue that that is.

The reason Asaf showed up the other day is because he accidentally clicked the wrong button : D

The plot thickens!

Moar plots =)

perhaps ymar could sell algebraic burgers. he'd have no competition.

@KannappanSampath May remind you of your promise to look at that thing? :-)

It is 23:00.

@DavidWheeler And no clients either. :-D

@JasperLoy Use alltheplots!

=)

Use matplotlib!

I can not make gnuplot ork on this computer =(

Write your own in C.

@ymar...nonsense! think about your special "axiom of choice" sauce!