Can anyone tell me why a horizontal dotted line appears across the chat at apparently random times?

@JonasTeuwen Ello : )

@MattN. Ello!

@OldJohn It appears when an important user wants to talk.

@OldJohn Not sure but I've thought it appears after no one has said anything for a while.

@JonasTeuwen Ok, can you tell me why we have to have compact groups to do Fourier series on them? I assume it has to do with the inner product (=integral)

if you switch tabs out of chat, discussion happens, and then you switch back onto the chat tab, the dotted line tells you where you left off

@RagibZaman Sounds more likely :)

@anon if you got answer or ... found any material on the related subject ... please tag me here. and thank you!!

@anon Ahh - makes sense

@anon I starred. I asked the same question and someone else might ask it again in the future.

@MattN. I think locally compact is enough.

@MattN. Oh so you are moving into abstract harmonic analysis eh?

@OldJohn You mean what I said doesn't make sense?

@MattN. What measures do you have?

@JonasTeuwen Haar ones. What else can I have? But at the moment it's just Lebesgue because Haar is the same on $S^1$.

@MattN. Good. When you integrate, you love Fubini right?

@Gigili Not exactly :) ... just thought the other possibilities might be more likely

@JonasTeuwen Sort of. Not sure where this comes in here. Where do we have a double integral?

@RagibZaman For existence of Haar measure. True dat.

@MattN. Often. Say in Plancherel.

@MattN. The point I am trying to make, your Haar measure does not necessarily have to be $\sigma$-finite.

But then your underlying structure makes it so that the functions you try to integrate actually simulate this.

@JonasTeuwen ok, my first question has been answered: we need to make assumptions such that existence of Haar measure can be proved.

May I ask another one?

@MattN. Mm, many reasons, this reason is purely computational to make it useful.

@MattN. Shoot.

Oops, I just answered it myself. Give me a minute to put question 3 into words.

@JonasTeuwen Ok. So we can do Fourier series of what functions? My first thought was $L^2$ because we need the inner product. But we cannot express all of $L^2$ in terms of characters because characters are continuous. So the answer is: continuous $L^2$ functions?

@MattN. First question: what convergence do you want?

@JonasTeuwen I don't know what choices I have. Let's start with pointwise perhaps.

@MattN. That is a very hard question.

In the first lemma here, why is finitely generated necessary? In particular I don't know what "unions of chains" means; surely unions of subspaces need not be subspaces, and unions of chains need not be chains (even with finite orders).

pointwise is usually too hard, $L^2$ convergence is often more "natural"

I know some cool theorems for the pointwise one.

Even in concrete fourier analysis on the reals, pointwise convergence of fourier series is harder to address than $L^2$

@MattN. What you want to do is prove that your characters form a basis for $L^2$, then you have your convergence.

@RagibZaman Understatement. Do you like Lacey-Thiele proof?

@JonasTeuwen Of Careleson's theorem?

Carleson's*

@JonasTeuwen So the answer I should've given is $L^2$ convergence of the Fourier series?

As you can tell, I'm not all too sure what Lacey-Thiele proofs are =P I know a little bit of fourier/harmonic analysis but no where near your level lol.

@MattN. Yep.

@RagibZaman Yes.

@RagibZaman Hmm, do you know the proof of Carleson's theorem? Lacey-Thiele provide a framework for this proof.

@RagibZaman Anyway, check out the paper. It rocks (ArXiv has extended version).

In class we only covered the statement and the lecturer told us it was first proven in the 60's and was famous for being a VERY hard proof lol.

@MattN. In that case it is a question about a Schauder basis.

@RagibZaman The proof is not so hard...

Maybe not the new ones, but the original perhaps at least to my lecturer.

people.math.gatech.edu/~lacey/research/carleson.pdf

It must be that paper

@JonasTeuwen I'm still thinking.

@RagibZaman Yes, check ArXiv.

@MattN. Go ahead :-). Do you know what a Schauder basis is?

@JonasTeuwen I think so. It's a collection of $f_n$ such that for an $f$ in your space you have $\|f_n - f\| \to 0$.

@MattN. Mmm... na it is more.

The other one, Hamel, is what lets you write every $f$ in the space as linear combination of $f_n$ in the Hamel basis.

@MattN. Schauder is the "infinite" version.

@JonasTeuwen I think $\|f_n - f\| \to 0$ entails infinite. But perhaps I'm missing a sum.

@MattN. Yep.

@MattN. If $(f_n)$ is a basis for $L^2$ this means that every $f$ can be written as $f = \sum c_n f_n$ where the convergence is in $L^2$. Now compare with your Fourier series.

@JonasTeuwen Right. I should've written $\| \sum_{k=0}^n a_k f_k - f \| \xrightarrow{n \to \infty} 0$

@MattN. Yep.

Well yes. We claim that the characters form a Schauder basis for $L^2$.

@MattN. Precisely. So if you prove that you are done.

But I have a problem with that: characters are continuous hence their sum is. $f \in L^2$ might not be continuous. It makes me think that we can find $f \in L^2$ such that we cannot find a sequence of characters converging to it.

@MattN. How do you define them?

@MattN. You have an infinite sum eh. Things get odd there.

@JonasTeuwen A character of $G$ is a continuous homo $G \to S^1$.

@MattN. $x^n$ is pretty continuous eh?

@JonasTeuwen Yes I know that actually. But it still disturbs me.

@JonasTeuwen Yes, quite.

@MattN. No, let $x \in [0, 1]$ and send $n$ to $\infty$.

Yes right.

@MattN. Still disturbing?

@JonasTeuwen Well then how about this: I want uniform convergence for my Fourier series.

@MattN. First try $L^2$ eh?

It is the easiest one.

Ok.

@MattN. What book are you readin'?

@JonasTeuwen Well that's a bit unfortunate: none. I'm trying to make do with hermitsy's lecture notes.

@MattN. What are you trying to learn?

@JonasTeuwen Everything.

Harmonic analysis on groups?

: D

@MattN. Holy cow.

I cannot help you with that.

But I want to know it all too.

@JonasTeuwen For now I'll just learn chapters 2-7. Currently looking at 3.

@MattN. Yep, 2 is crucial (and actually not so hard, don't be scared).

@JonasTeuwen Done most of it already anyway. Almost.

@MattN. Then you would have immediately cooked up the Schauder basis! :-). Retry!

@JonasTeuwen What does it mean?

@JonasTeuwen For now I quit worrying. I'm back on these for now.

@MattN. If you would have read it you would have known the answer to the $L^2$ question, so try again and so some exercises to absorb the material better.

@MattN. Ah!!

@MattN. Kickass.

@JonasTeuwen I'm not finished, I said almost!

@MattN. Take more.

@JonasTeuwen Which question (of the many)?

@JonasTeuwen I think he doesn't prove that the characters are complete (i.e. orthonormal basis of $L^2$) he just assumes it.

@MattN. It is a standard result.

@JonasTeuwen So the answer is: I can do Fourier series for $L^2$.

@MattN. Yep.

Because we assumed that characters of $S^1$ are complete in $L^2$.

Great.

He really should write notes with care.

Hello

Is the open $n$-cube usually denoted $(0,1)^n$?

@JonasTeuwen Is this the question you said I would've known the answer if I'd read chapter 2?

@MattN. Yes.

Ok.

Hey, thanks bro! I think I know more now.

You are awesome!

Also, you were awarded the [counselor] badge the other day : )

I should go to sleep.

Thanks for today Jonas!

Good night!

Hi @Alex.

Hi, I've got a problem relating to categories, but it's kind of messy...

so... is it ok to ask it there ?

It is.

Ok... I have to say beforehand that it's not a textbook problem of any sort, more like a free experiment I'm working on... also, I'm not very good at math (I'm a chemist :) so I apologize in advance if things get really messy

@MattN. Thanks :-). The what badge?

@MattN. Good night!

@JonasTeuwen I think Matt might have invented that badge - did you give him some counselling advice recently?

@OldJohn Yep. I was already searching through my badges :[.

@JonasTeuwen searching through my badges doesn't take long at all!

@OldJohn My eyes sometimes are not so good, so I have to look very carefully :-).

@JonasTeuwen Your eyes are several decades younger than mine!

@OldJohn Hmm, I don't know what it is but often my peripheral sight is much "larger" as in the part where I can see sharp is smaller.

@JonasTeuwen Hmm sounds like inverse glaucoma :P

(I have the direct version)

@OldJohn No, like glaucoma actually! But I do not have it. The ophthalmologist said I do not.

@OldJohn You have glaucoma? Sucks.

With glaucoma, the peripheral vision is very poor - I only have a very mild version, and have it under control, fortunately

@OldJohn Yes, that is what I mean. The peripheral vision is "larger" as in the region where it is sharp is "smaller" :-).

Also I see black "noise".

@JonasTeuwen Black "noise" sounds a bit like some of the symptoms of migraine - I used to get "zig-zags", but they seem to have gone now :)

Is the open $n$-cube usually denoted by $(0,1)^n$?

@PeterTamaroff I would denote it like that (but I am not an expert)

@OldJohn But... no pain! :-).

I want to figure out where all my health problems come from, but as yet I have not found it.

@PeterTamaroff The unit cube, yes.

@JonasTeuwen Yep - I never got pain with the "zig-zags" - just weird vision for a while

@JonasTeuwen Dope-

Have you guys seen the news about the killing in the Cinema in Denver?

Yes.

Fucked up.

I'll keep downloading my movies.

@OldJohn What causes the zig-zags? What do you see?

@OldJohn The migraine zig-zags?

@Gigili: ok the problem I have starts with a result I find quite interesting: the automorphism group of objects in a monoidal category freely generated by an object $A$ and a morphism $A \to A \ocirc A$ is the Thompson group F

sorry, an isomorphism

@AlexPof Oh, dear. I know nothing about that. Perhaps other people can help you out. @John, for example.

But there is no question?

@JonasTeuwen I never found out what caused them - they happened at totally random times. What I saw was sort of zig-zag blurry lines just outside my central field of view, and they alway faded after about 30 minutes

and they only happened once every few weeks or months - very odd

@Gigili: aw sorry about that. Maybe I'll try to explain quickly what I'd like to do, maybe it'll sound familiar ?

@OldJohn I got one of those shimmering rainbow zigzags for the first time a few days ago. At first I was concerned till I googled it and realized it is fairly common, usually associated with migraines. I didn't have one. In fact I rarely get headaches. It went away in about 5 minutes, but was very annoying while it was there.

@BillDubuque Sounds just like what I had - no headache, just weird rainbows, and faded after a while - seems to be nothing to worry about

@BillDubuque Had one once while driving down a motorway - just stopped and had a coffee, then carried one when it went

@OldJohn I supposes a few decades of staring at CRTs and LCDs and books will eventually take some toll

@BillDubuque Yep - I am sure

@AlexPof Yeah, do it. But I don't think I can be any of help. Perhaps if you explain the problem other users will be able to help too.

We need to get out more!!!

@OldJohn Persistent migraine aura?

I have it such that I have lots of trouble seeing things for half of the day. I can hardly use a computer...

@OldJohn Perhaps they are also caused by staring at extreme editing, of which I've been doing too much as of late...

@Gigili: ok, I'll give it a try :)

@JonasTeuwen sounds about right

@BillDubuque not my editing, I hope!

@OldJohn Did you used to have lots of migraine?

@OldJohn We have one user that was doing some very extreme editing ... pushing the boundaries on many fronts.

@BillDubuque Oh, if you don't have it regularly it is indeed not so uncommon. Many people have migraine at least once in their lifetime and it can surely happen without the headache. In that case you are very lucky as it pretty painful, I can tell you.

@JonasTeuwen Very rarely - just the vision thing, almost never headaches - I have been remarkably healthy considering my lifestyle (once had 5 years without any days off at all)

So I was thinking about the case where this isomorphism is just a morphism, and somebody explained to me that one could always take the localization of that category.

@BillDubuque Ahhh - MK

@OldJohn Pretty cool :-). I have only been very recently quite ill, but I have lots of small health problems :-). Like this...

@JonasTeuwen I don't believe I've ever had a migraine, but dealing with cranks on sci.math probably comes close!

They should be related somehow, it would be pretty odd to have them all independently I guess.

@BillDubuque Well, if you would have, you would probably have known.

It is usually a very painful unilateral headache, the unilateralness is usually enough to know it is migraine.

Haha... Didn't they adapt the John Baez Crackpot index?

I then tried to apply it in the case where objects are sets, for example when A is $\mathbb{Q}$, or the dyadic rationals. I found an injective map $A \to A \times A$, but then I ran into some cases where group elements from F don't seem to act on these sets. So I'm wondering what's wrong...

@JonasTeuwen I don't think one could ever hope to categorize the motley species of cranks inhabiting sci.math. We even had a real mathematician who had crackpot physics theories about blowing up the moon, Alexander Abian. But his math posts were great - he was a very good teacher.

@BillDubuque Is the group still so active with many good forums?

@BillDubuque Abian sounds like a fun guy - more interesting than most of the cranks I have come across

@JonasTeuwen When MSE went public I coaxed most of the prolific noncranks to come here. Now it is mostly cranks and those who like to bash them, combined with those, for some strange reason, who don't want to leave (or don't know how). At one time the quality was comparable to here, if not slightly better (since there was not much competition).

@BillDubuque Hmm... sci.math.cranks?

@JonasTeuwen Yeah, but "cranks" is redundant now, alas.

A good crank can be fun. If he keeps being original.

@BillDubuque The physics cranks are the worst of course, here you have a nice summary of people telling a Nobel prize winner in physics he does not understand relativity.

@JonasTeuwen I interacted some with Archie Pu, trying to see if I could get him to see the light. Alas, I paid a price. Later he started posting many crazy threads like John Conway and I can't prove XYZ or "errors of so-and-so". Stupid Google indexes them high because they are titles of threads, so at one time if you googled my name as a layperson you might be misled!

@BillDubuque Well, actually... he would make you more "findable" that way 8-).

Once attended a lecture course by John Conway - brilliant, but the exam at the end was impossible :-(

@BillDubuque Well, yes... can't google remove those from the index? Probably they won't.

@OldJohn Which John Conway, the complex one or the surreal one?

@BillDubuque It was a third year course in number theory - back in the early 1970's

Ahh - sorry - it was the surreal one

@OldJohn Probably the surreal one, i.e. John Horton Conway. At Princeton?

That was the guy - but before he went to Princeton

@OldJohn Where was he then?

@BillDubuque Cambridge UK

@OldJohn Ah, yes. Are you on that side of the pond still?

@BillDubuque Yep

@BillDubuque Haha check out this: arxiv.org/pdf/physics/0403092v1.pdf Perfect. I wonder... how can such persons write reasonably eloquently and still be complete morons?

@OldJohn JHC used to be active in various eforums, e.g. math-history-list, but apparently no more. I miss interacting with him.

@OldJohn Do you know the answer to Alex's question?

@BillDubuque he was a fantastic lecturer (back when I heard him) - enthusiasm and wit and some very smart maths very came easily

@Gigili which question?

(did it mention categories?)

@JonasTeuwen Reminds me a bit of Archie Pu (Archimedes Plutonium). Luckily he hasn't discovered us yet, like WM has.

@Gigili Sorry - I don't think I can be any help there :(

@JonasTeuwen Strangely enough though, I did have some success at one point teaching Archie Pu a little about proofs by contradiction (Euclid's proof of infinitely many primes) Unfortunately he then got stuck on another point, thinking that proofs alternative to his must all be wrong (or something like that). Proofs by contradiction are highly confusing to laypersons (even to some math students).

Umm, sorry @AlexPof. seems we can't help you. Why don't you ask on main?

@BillDubuque Sure, it is confusing to math students because it is not a common thing you would do in daily life. But... why do you spend your time on this? :-). You know the statement about making something "foolproof", right? Even if you fixed one of the fool errors, the fool will find a better one.

But you do need some guts to tell a Nobel prize winner in physics he does not understand physics :-).

@JonasTeuwen I'm very interested in understanding the source of logical errors, even in cranks.

@BillDubuque Mental illness...?

For reasonable people this is a good thing to do, you can probably learn from it as a teacher. But for people who lack the possibility of rational reasoning...?

@JonasTeuwen For some, but they are basically wired the same way by evolution, so the errors are often similar. It's interesting to study them, for the same reason that neuroscientists study brain-damaged patients and other anamolies (cf. popular expositions by Oliver Sacks).

@Gigili: I actually did, but I tried to divide it into separate, clearer questions. With little success however, since I probably don't know enough math to make it clear enough...

@BillDubuque Well... as long as you're having fun it is okay I guess :-).

@JonasTeuwen I've always been fascinated by the way the mind works. If I wasn't a mathematician I'd probably be a cognitive scientist.

@BillDubuque I wonder... would there have been cognitive science research on this phenomenon? Probably cranks don't want to join a "crank research" as they think the other ones are cranks. Perhaps one can tell them they want to compare them with cranks. Would be pretty cool. They seem to be quite intelligent but still... they write nonsense!

It is somewhat like the Dunning-Kruger effect.

Maybe it is the Dunning-Kruger effect, but additionally their verbal skills are less damaged.

@JonasTeuwen I'm not aware of any such study on cranks. Yes, I agree that there are similariities to DK, somewhat local vs. global crankiness.

Would actually be pretty cool. I'll check the list of SE sites, perhaps there is some cognition related one. Maybe someone is interested in researching something like this... It is really peculiar. They seem to know many definitions and books but... fail to understand them at all.

@JonasTeuwen Similar to many of our drive-by questioners, alas!

@BillDubuque Oh well, the quality is still quite good.

Aha WM - Ross Millikan has caught one of my (mathematical) typos ... and corrected it without fuss

@OldJohn Good, right?

@JonasTeuwen Yes - excellent

he must have been looking at the maths and not just the presentation and words :)

@OldJohn Do you feel "mathematically challenged" now? I do.

@JonasTeuwen I have always felt mathematically challenged - but I am having fun now that I have retired and can do whatever maths I like ... as much or as little as I like

@OldJohn Does that imply that math before retirement wasn't fun?

@BillDubuque Oh I think it has always been fun! - just that now it has more freedom :)

@OldJohn Ah, I see. See the Cantor quote in my profile..

@BillDubuque Ah yes - but where does the other quote come from? I like it even more.

@OldJohn It's a famous aviation quote that I morphed to math. It is often attributed to Da Vinci, but apparently this is incorrect, see here.

@BillDubuque Fascinating!

@BillDubuque @JonasTeuwen I must get some sleep - G'night folks

@OldJohn Good night! :-).

@OldJohn Sweet dreams of flight...

:))))

Considering how much the lot of us has been getting all riled up in meta, it's interesting that this just came up...

@J.M. That's basically the same as I've been stressing on meta for a long time about the dangers of unwelcoming behavior. He mentions 3-4 years but we're already seeing this at 2 years (and even earlier). We need to shape up or risk becoming stagnant.

@JM Hi! :-).

@JonasTeuwen I wonder why there's no medical SE forum

@BillDubuque Because most people actually are not competent enough to value those answers I guess.

If there is a thing that is full of cranks it must be medicine.

Even worse, full of people who love logical fallacies like: You are not a doctor, so you cannot know anything about this.

Apparently an education is a proof of sanity then. So my physics paper is also wrong... which has been accepted 8-).

@BillDubuque People would die.

@JonasTeuwen Sure, but every field has its cranks. Perhaps SE is worried about legal ramifications.

@BillDubuque It is much more harmful in medicine. Sometimes people can have quite serious complaints and when they follow a crank things can become much worse.

@JonasTeuwen But there are already many other websites that handle medical questions. Would you expect better advice here than one of them?

There are medical forums online with real MDs answering the answers, and usually it contains something like: go see your GP.

Yes, but SE is quite big, so if something goes wrong because of a crank it might be very well possible that they get legal problems.

I would expect better advice, but SE is open to anybody.

@JonasTeuwen I think the real problem is getting voting to signify something meaningful. We have the same problem here. Usually the highest voted answer is not the best, but rather the quickest and/or most elementary. However, if one had real doctors voting, that could be interesting. One could distinguish between votes from doctors vs, patients, to get feedback both ways.

Hmm, the thing is also that other sites like MSE enable people to get access to expert opinions. For these things a medical SE could cover people actually already have access to somebody with such knowledge: their GP.