@tb I'm not as well )

Well, I saw it from far away when I was at an open air about ten years ago.

their? or just another fireworks?

@tb Fireworks in the sky are wonderful. On stage, not so much.

@Gortaur yes, theirs.

@robjohn: I'm a bit schizophrenic here. I like the look of it, but the old ecologic activist inside of me spoils the fun somehow.

or were you speaking of natural fireworks? meteors and the like?

@tb poor you, so you cannot enjoy'em in the full power

that's a pitty

as for me I don't care about fireworks at all

@tb I have seen fireworks once a year every year on the 4th of July (Independence Day in the US) so I guess I am simply brainwashed. But there is nothing better than a good meteor shower.

I have a question

@Gortaur: shoot! is it about a typo?

Go to MSE... oh, wait.

in Russian notation it's called a condensator: compact set A and B\subset A is open

you would like to find a function f such that

f = 0 on A complement

f = 1 on B

int |\nabla f|^2 dx -> min

So you're working on R^n?

oh, yeah

f have to be smooth

at least in 2D the solution is a harmonic funcion on A \ B

the minimum value of the last integral is called the 'condensator capacity'

and the pair (A,B) is called a condensator

I think it is harmonic in any dimension.

@robjohn: I distinctly remember the peak of the Perseides in the early nineties. We were traveling with an old VW bus through the mountains and arranged it so that we were on mountain passes during the night.

@robjohn Euler-Lagrange equation?

yeah, or any other Calc of Variations method

aha

@t.b.: there was a meteor shower visible in Europe last week that was supposedly 100+ per hour

anyway, have you heard about that notions? Like haveing their own names and so on

@robjohn: I know but it was very cloudy and I couldn't go out of town.

maybe I should also ask Byron

@Gortaur what notions?

condensator and its capacity. in the way I've just described

you want a name for the problem?

yes

I think integral to be minimized is the Dirchlet one.

one of the Dirichlet ones )

I mean in Russian terminology it is 'capacity of the condensator'. I wonder if there is an equivalent problem known outside of Russia )

Hi all from a beginner

@AkramHassan: hi. what time is it now?

@AkramHassan greetings.

i fell in love with mathematics 5 years after graduation :)

Did we scare off Ramana?

@robjohn no, she just enjoys a new book I guess

@AkramHassan High school or college?

graduate engineer

@robjohn so you don't know the answer on my question, right?

@Gortaur I don't off the top of my head. I was looking for some references.

it's 1:36 PM here in Egypt @Gortaur

Dear Rob, thank you so much - but I mean that if you don't know it like 'hop, that's it'

@AkramHassan Do you have a particular field you are interested in?

don't waste your nighttime on it, please )

I really don't want to bother you so much

but it is an interesting question.

@AkramHassan thans, so I don't need to add a new clock )

lol , where are located @Gortaur ?

We are located in the City of Leiden, the Kingdom of The Netherlands

not so far from the Palace of Her Majesty

@Gortaur: I didn't add a new clock, just put him under Asaf's timezone.

@tb: I've just answere to your comment about Stokes formula in the course of Analysis

@robjohn do you mean that I'm also in Asaf's timezone, not Asaf is in mine? ;|

yes @robjohn , still discovering , so far i am more attached to computational geometry . but i am still looking for an area where i can find (visual problem) , you know what i mean ?

@JM hi. wanted to ask you - is it a fly on your gravatar?

@Gortaur I have you guys on two clocks, before I realized they were in the same zone

@robjohn my pleasure )

you have a particular research area @Gortaur ?

@Gortaur He described it in detail...

@AkramHassan: I do. it is stochastic processes

@robjohn but he never mention if it is a fly

@Gortaur I don't know about the terminology either. However, it should be an application of the usual Dirichlet principle (minimize the Dirichlet energy on A \ B with the appropriate boundary values).

@robjohn Now that's one good fake! :D

@AkramHassan Do you know about Soddy's theorem then?

@tb thank you, Theo - maybe I'm searching in the wrong books

nope :(

i am still a beginner @robjohn

@JM I thought it looked very similar

yesterday we were so tired that we refused the zoo trip and just went in the evening to check out if Dutch pubs are still ok

@Gortaur: btw. I wasn't implying that you were complaining on Michael's div, curl, grad question. I was just reminded of the genesis of Bourbaki

zoo vs booze

booze has zoo in it :-)

@tb I see. I hope I haven't gave you an offence

@robjohn , what's research your area ?

Set theory, but I'm not robjohn.

@Gortaur no, not at all :) don't be so timid. I can take it.

@robjohn how can you think in that way, Sir! I was with the Lady

@AkramHassan I don't even know if Soddy's theorem is considered computational geometry. What is covered in computational geometry?

Soddy as in those circles?

computing the boundaries of a convex hull for instance @robjohn

@AkramHassan analysis (harmonic), though I dabble in combinatorics, and other areas.

like what robjohn ?

@robjohn: I need that link to that software again.

@AkramHassan Ah. Soddy's theorem (and Gosper's extension thereto) is about mutually tangent spheres in R^n

@JM indeed

@AsafKaragila logiclx.humnet.ucla.edu

@Gortaur , stochastic processes is still too advanced for me :) . what is it like ?

IIRC Soddy managed it without the machinery of coordinates. Of course, with coordinates, establishing it is somewhat easier.

@JM I have no idea. Do you know how Soddy proved it? I proved it and a new related result a couple of years ago.

@robjohn , computing the boundaries of a convex hull is an example of computing geometry

@rob: Is there a place where I can see this a demonstration of the software?

@rob: I lost my copy of that paper, sorry. All I remember is Soddy's was purely synthetic.

@tb I will do my best

@AsafKaragila you can just download it and try it out. There is documentation on how to use it. If you have questions, just ask.

@AkramHassan like this: en.wikipedia.org/wiki/Martingale_(probability_theory)

@JM but I used coordinates and matrices.

@robjohn , @Gortaur , it's a pleasure to meet you guys :)

So a Martingale is not a bird? Jeez! :-)

@AkramHassan nice to meet you, too. You leaving already?

I feel rather silly...

@rob: I always did say, there's a time to use an axe, and there's a time to use a buzzsaw... ;)

Soddy's was rather long...

Have you been walking today, @tb?

@tb what, Georges answer?

@robjohn: Well, I missed an n in the question and I didn't dare doing what Georges did...

@JM Mine is about a page or two.

@rob: In that case, a buzzsaw is indeed justified. ;)

@JM coodinates and matrices is a buzzsaw?

can one upload PDFs somewhere to SE?

Well, I would say part of the reason why Descartes came up with coordinate geometry is that it eases the proof of synthetic facts...

@JM what do you classify as a synthetic fact?

(At the very least, to take plane curves as the example close to my heart, establishing properties synthetically is doable, but long. Coordinate geometry makes things a lot simpler.)

@robjohn Nope, you need a filehost like Sendspace or iFile.it .

Are you talking about Axiomatic Geometrical proofs?

@robjohn: nope

@rob: Yeah, like those. On the other hand, you do need a different set of axioms/postulates for coordinate geometry...

For instance, in the treatment I'm accustomed to, the Pythagorean theorem is pretty much a given.

@JM I'd never heard it referred to as synthetic proof.

or whatever.

The definition of "synthetic" I'm accustomed to is "what Euclid, Apollonius, and cohorts would have done".

@J.M.: your gravatar is staring at me... :-)

Maybe "coordinate-free" is a less loaded term...

it looks like a bird face.

@rob: It seems that I am installing logic in a .jar ;-)

Wait, it's Java?

@AsafKaragila what OS are you using?

@JM yes

Windows XP, sadly.

@AsafKaragila It works there just fine

It's the department's computer. Old old.

Why do I need to update after downloading the recent version? It does not make any sense.

The Mac installation is a bit nicer. It is all in an app bundle on the Mac.

In my book, XP is "old". 98 is "old old". 3.1 is "somewhat dusty".

What about Dos 3.22?

Or Linux 0.1?

What about BeOS 5 Personal?

The version on the download site is not the latest. We update frequently and some institutions require a modified version. so it will usually update the first time after installing.

@AsafKaragila :-p

The Mac installation can be made to work under Linux/Unix.

Hmmmm. It seems that this software is not too useful for this course. Perhaps somewhere else.

You have to use a command line to start it though

I fear not the command line.

In fact, I always have a running prompt or three at home.

@AsafKaragila you only need the command line using Linux or Unix though

Yeah, I figured as much.

I use Unix, MacOS, and Windows XP on my laptop.

@all : I sometimes wonder why do we usually measure the mean square error between functions in general real world problems ?

because the theory is relatively well-established and nice.

so much, even when other measures of error are more appropriate

is it because it is analytically tractable as opposed to for example mean absolute error ?

That's one.

Optimizing with that measure is easier, since the underlying function is differentiable.

this will also lead to inner product as a similarity measure between functions ...am i right ?

With, say, the Manhattan norm (your "mean absolute error"), the machinery needed is more intricate.

yes

and the orthogonal expansion too

orthogonal basis expansion leads to minimize mean square error

Of course, if you want to do things properly, optimize with respect to the Euclidean norm first, and use that as a starting point for optimizing with respect to the desired norm.

@robjohn it is, but I guess not in English (

@J.M. : in some neat real world applications, information is present generally as a physical quantity varying with some independent parameter, say for example time (it could be anything).......and people genreally tend to represent it with a function ....i would like know some things regarding this representation

@AsafKaragila Are you using xmonad already?

I would say that one really needs to build a good theory, first and foremost...

@JonasTeuwen No. I like awesome.

only then do you worry about the functions.

yes ofcourse....but i am just taking a general idea on existing theories

@AsafKaragila That is also fine.

say for example an audio signal...it is a variation of acoustic pressure with time captured by microphone as an electric voltage or current

@JonasTeuwen I've been using awesome for a long time now. Since GNOME 3 came out.

Haha. Yeah, I can understand that...

@Rajesh if something varies with a parameter then that something is a function of that parameter by definition of depending on a parameter and by definition of a function

@t.b. : Yes

So what's wrong with representing "real world things" that depend on parameters by functions?

i meant to mention a particular case of real valued functions of a real parameter....say an audio signal as a variation of acoustic pressure with time.......all i am trying is to give present context.....i am not done with it yet

nothin wrong !

Anyway, I was going to tell a little bit more about my project. The Hardy Space H^1 turns out to be quite useful in the Euclidean theory for example in PDEs. Now many of those results can also be useful in SPDEs if we have a different measure, in this case that is the Gaussian measure.

The Gaussian measure is non-doubling so that messes everything up.

@Rajesh: I like to think of a function as something that depends on a real variable as an "analog signal"

@JonasTeuwen Stein and co use H^1 a lot.

yes

So, I'm trying to figure out what an appropriate Hardy space would be in this case. Certainly we have to replace the Laplacian by something else since that is not symmetric. It turns out that the Ornstein-Uhlenbeck operator works.

@robjohn Yes I know :).

And then we of course want to know if the dual is BMO (or what is BMO in this case?) and if the Riesz transform is bounded and so on.

@t.b. : i actually started of with the use of mean square error

@Rajesh: I saw that. That's where I'm heading.

you may seem some lines of conversation between me and J.M. just a few minitues ago

ok

Let's stick to audio signals.

ok...minimizing a mean square error between functions generally preserves the shape of the function

am i right or there are some inccuracies in here

you have many ways of measuring the "error" between functions

the mean square one is particularly convenient for many reasons

Well, just to take a random example, you'd get a poor approximation if you attempt to approximate an exponential with sines and cosines...

yes

So if you're approximating something you see in the real world, the onus is on you to choose a good basis.

generally approximating in the sense of mean square error orthogonal basis expansion

(How to pick a good basis is a different can of worms altogether.)

sorry if am to slow in getting to the point and wasting time

Rajesh, the orthogonal basis is not a given

you are free to choose it according to your ends and needs

What t.b. said.

you mean we can choose any orthogonal basis ?

One application might demand a Fourier expansion, another might demand a Chebyshev expansion or a Hankel expansion... it depends.

yes.

speaking of audio signals, we can for example choose sine and cosine waves because they are easy to produce

that yields the usual basis you use in the Fourier approximation

We could also choose square waves if we please, and that would give the Haar basis

If the signal has special symmetries, you might get away with using sines only, or cosines only.

(So my question on the preduals can turn out to be useful because if I cannot find the dual of "my"\ H^1 maybe I can find the preduals of "my" BMO)

Like if the function is even or odd.

yes.....i want to mention first is that approximating in the sense of mean square error generally tries to preserve the shape of the function....let me know if am getting any thing wrong here

It depends on the application you have in mind which one is the most appropriate.

yes

Mean square error = L^2-error?

yes

@jonas yes

@Jonas, yep, 2-norm or Euclidean norm.

Aha.

@Jonas: I think I missed, what are your H^1 and BMO spaces?

in audio we generally use Fourier expansion....

@Jonas L^2 convergence used to be denoted by f = l.i.m. f_n (limit in the mean) by Riesz and company

@Rajesh: yes, can you express in your words why?

the point i'd like to make is that in audio the use of Fourier exapnsion and the like is that the audio information (what can be heard) is present in the shape of the audio signal

that was'nt an answer to your previous comment

i'll try to answer it soon

@robjohn I didn't mention them. Let me find an image.

@tb: Now I'm adding bibliography to other people's posts too... shame on you :-)

That isn't so bad a habit to acquire, Asaf.

@Asaf: I'm proud of you :)

@AsafKaragila some people... they just spend too long on other people's posts. :-)

:P

@robjohn: and get headaches ? -_-

@tb indeed

@t.B : it is because of the perception.....human ears tend to distinguish sounds as generally high frequency or low frequencies....there are more intricacies with them...i am not intending to get into them

i.imgur.com/mUAK1.png Here. h^1 are "my" Hardy spaces.

@AsafKaragila Will you add bibliographies to my posts as well?

@JonasTeuwen In the definition of the maximal operator T*, is the L a constant (I hope)

No, that is the Ornstein-Uhlenbeck operator and e^{t^2 L} is the semigroup generated by that operator.

@Rajesh: a very rough model would be that we can detect signals depending on their frequencies. So if we have a pure 440 Hz signal there is exactly one detector stimulated, so it makes sense to decompose an audio signal according to its frequencies.

- 1/2 sum D_i^* D_i

okay.

@t.b. : i agree with you...it depend on the detection machinery we have and the medium of transmission

@JonasTeuwen I didn't see any other reference, so I wasn't sure.

Yes, I should have mentioned that, but I just copied a part of my introduction. This reminds me that I should also explain what the L is there...

The philosophy with the method of using a mean square error (or the likes of Lp norms) is essentially under the assumption that the information of the signal is present in the shape of the function representing the signal...this is what happening in almost all of our real world applications....barring some which i am not aware of

@t.b. :

@Jonas: I worked with H^1 and BMO as Stein was my advisor. Never heard of the Ornstein-Uhlenbeck operator, though.

Cool!

Yes, SPDEs are not that well known.

(Stochastic)

@tb :-)

Didn't Stein retire this year?

@Rajesh: if we have a complicated signal and we want to determine the contribution of the pure A sound to that signal, that corresponds to projecting the function representing the signal orthogonally (with respect to the L^2-product) onto the 1-dimensional space generated by the function representing the pure A sound

@Jonas: I worked with psedo differential operators, too, and they are not all that well known either.

@t.b. : yes

Oh :) I have followed a course on PDO where they have used the AMS book by Alinhac & Gerard.

But consider the case we want to measure the similarity of two signals : we just take an inner product

...and what measure are you using in that inner product?

@Rajesh: The point I was trying to make is that viewing it this way, using the mean square error is the natural thing to do (and JM beat me to it, yet again)

i agree...

I'm sorry! I don't mean to t.b....

:D

:-)

@robjohn: If you have worked with BMO, would you know if there exists a characterisation for the preduals of BMO? (I have posted a question about that)

Getting back to the math: certainly you take an inner product of some sort. On the other hand, there are quite a number of inner products, with different weights, measures, and whatnot.

@JonasTeuwen not right away.

Okay.

@Rajesh: But that's just another instance of the same thing: <f,g> represents the contribution of g to f or of f to g

so it measures the similarity

yes

let me type what i'd like to say at this point

@Jonas: this paper came out soon after I was at Princeton. I don't know if it will help at all.

@tb In a very unidirectional sense.

@robjohn: definitely

@t.b. , @J.M. : I'd like to consider a case where the information is present in the signal not just in the form of the shape of the function representing it but also present in the manner in which the function is behaving............it can apply to any signal or any function used in any application........the info the function is carrying is not just in its shape but also in its behaviour.....stating precisely..........shape is not all that matters...the behaviour also matters.........contd..

it also means that. taking smooth approximation to such functions (Stone-Weierstrass approximation) is not going to be good enough..........do you think its a cool idea....especially if i am able to influence real world applications (could be anything including Physics)

it been hovering in mind for the past one year

If you mean approximating with functions that aren't "smooth", that's what wavelets do.

no

why not?

Wavelets also approximate the mean squuare error...trying to preserve the shape of the function

I went with your "smooth approximations aren't good enough" bit.

i want to eradicate this type of approximation

I don't understand what you want. What is the "shape" of a function?

In using the mean square approximation we tend to preserve the "shape" of the function

@JM Harr wavelets for example.

thats what i intuitively mean

@rob: Yep, a whole family: Haar, Daubechies, those Mexican hats... I've lost track myself. :D

@RajeshD do you have a formalization of what you mean by the "behavior" of a function?

yes

Apparently except for Rajesh, none of us know the difference between "shape" and "behavior"...

plz hold on

@JM If I misbehaved, I know that my parents would whip me back into shape. :-p

@robjohn : :-)

@rob: Mine would, too. Not the most pleasant experience. :D

@JM today we have so many titles that would need some cleaning up "mathematical induction usage" "obvious statement, how do we prove it", "how to solve this simple equation"

not very informative...

@t.b. Unfortunately for us, most people ask without thinking that other people might also be asking the same questions...

@tb that was a problem that was very bad on sci.math.

ses this class of function........mathoverflow.net/questions/61797/a-class-of-functions-closed : note in the comments by Pietro Majer (in his answer) that the set of rationals can be generalized with any countable dense subset. If we take a smooth approximation to this (stone-Weierstrass) we still loose information about the behaviour of the function which is quite important...........

ooh, a closed class of functions.

@robjohn: you're mean squared :p

is it cool

I'm hungry.

My blood sugar levels are dangerously low and I'm about to crash. Whoever is not a grad student, please send me some food.

Can't you get candy or something?

There should be a site like that, feedagradstudent.org or something. People could buy food for a hungry grad student.

So there will be pictures of emaciated 20-somethings?

@Rajesh: you can produce functions with all sorts of ugly behavior, after all, "most functions" are not differentiable anywhere

have to go afk for a while. bbl

See you, rob.

see you

@JM yeah

see you

@t.b. : oops is it ugly ?

The operative word is "most".

I didn't say your class was ugly. I was trying to point out that it is very rare for a general function to look anywhere near smooth

yes

Given the freedom that you have in producing functions mathematically, it is not surprising that you can concoct all sorts of behavior.

if a function were to carry any real world information embedded in its behavoiur also, then these class of functions seem to be the best cadidates for me....intuitively (not math in it) i this what i feel

I still don't understand what "behavior" is and how a function can have information embedded in this "behavior" you speak of.

by behaviour i mean differentiability at a point

what J.M. said. + I'm sorry but I fail to see why differentiability of a function depending on the denominator of a rational and smoothness at the irrationals should in any way be tied to the real world

@t.b. : the definition can be extended to any countable dense subset of (0,1).....not just rationals

sure

you can see about this in comments by Pietro in his answers

so why not.....

a countable dense subset seem to me as a good thing

yes, but again, getting control on a countable dense set is not much, as Pietro aptly showed

@t.b. : could you elaborate a bit

The first observation to make is that you can easily produce a function that is exactly n times differentiable at a single point and smooth everywhere else

@Rajesh: to add to t.b., you wouldn't happen to know about splines, would you?

Now do this for each point of your countable dense set. Scale your functions appropriately and combine them. That's what Pietro did.

i'd discard all function that does not belong to this class

in my theory if i were to produce

variation of any physical quantity would be represented only by a function of this class....

So a square wave doesn't exist?

yes

but a triangle wave does

there will not be place for any other function in this physical theory

no

why not?

even triangle wave doesn't

trignle was is smooth everywhere except at vertices

so a sine doesn't exist either?

yes

But there are circuits that produce square and triangle waves...

...and sines

@t.b. : I have not yet invented the theory

i wish to though

i'd like to prove they aren't what they look like

If you've taken out most of the usual periodic functions out of commission, then what in this green earth are you going to build your appproximants from?

@J.M. : I wish i had an answer..hope to find a way

they aren't that bad

the green world you see could be these class of function

i'll be back after a few minutes...stepping out for dinner

Above all, our theory explains why simplicity is so highly desirable. To understand this there is no need for us to assume a ‘principle of economy of thought’ or anything of the kind. Simple statements, if knowledge is our object, are to be prized more highly than less simple ones because they tell us more; because their empirical content is greater; and because they are better testable. Karl Popper, The Logic of Scientific Discovery (1959), 43. Simplicity and degree of falsifiability. P. 142

Well damn, I can't compete against Popper in the business of eloquence... :)

Couldn't resist.

In short: Ockham's razor.

exactly

@t.b. : I couldn't understand your quote

@J.M. : I think the question as to how i approximate or the way in which i approximate would be of first thing then comes what functions i would use

Wasn't that what t.b. and me were trying to impress earlier?

You need to pick your basis functions properly.

what JM said.

In addition, you're depriving yourself of all mathematical tools by focusing on your class of functions. You don't even have a vector space of functions, because the zero function does not belong to your class, you can't add, you cant subtract, etc

No use playing a game if you've given yourself every possible handicap, you know.

if the existing things aren't good .......there might be new things hidden somewhere.........anyway...i know there's no point in arguing when I don't really have anything solid in place....but i have some hope somewhere on these lines

I'm sure you have a good (but not necessarily satisfactorily answerable) question in there somewhere. You just need to figure out a precise formulation.

@J.M. : thanks for the words

@robjohn: Looks interesting, thanks!

hi @Sri

Oh hi @RajeshD

sorry if i had mispelled your name i thought first 3 letters are enough for the automatic alert