Too much Facebook for you, Sir.

in portuguese the word we use for Field is the translation for body.so prof usually make jokes about that

Yes. In Dutch too!

"Lichaam".

@PeterTamaroff Why? Facebook?

@OldJohn Do you know about the polynomials if the term "basis" also makes sense outside the vector module context?

@Charlie Yes, we use "cuerpo" too.

Like how you build the polynomials as linear combinations.

@JonasTeuwen nice!:D

@PeterTamaroff only in english it's not funny

@JonasTeuwen not sure I understand the question :(

@OldJohn An element from $K[X]$ the polynomial ring variable $X$ on the field $K$ are thingies of the form $\sum a_i X^i$ right?

If you would see it as a vector space you could say the $\{X_i\}$ are a basis. Is there such a thing for a ring as well?

Like as a generator.

@JonasTeuwen yes

Only like "generator"?

@JonasTeuwen a basis for a ring?

I am trying to find a way to "generalize" I mean abstract the phenomenon that if I have polynomials in one variable $t$ which have the obvious basis, I can make a basis for the polynomials in two variables $t$ and $\partial_t$ such that the new basis for the last polynomials is like just adding the partials!

Okay, I have $\mathbb Z[t]$. I also have $\mathbb Z[t, \partial_t]$. I want to find the algebraic way to say that $\{t^k \partial_t^k\}$ are a basis.

@JonasTeuwen are both of those exponents meant to be always the same (k)?

Yes.

Hmm.

Right, on a subclass of polynomials perhaps.

Let the second one be $k'$.

They no commute!

OK - then I suspect there should be a way

I figured the last one is a Weyl algebra! 8-).

hm - don't know what a Weyl algebra is :(

It is an Ore extension of $\mathbb Z[t]$! Not sure what that means.

@OldJohn A ring generated by $t, \partial_t$ :-)).

ok

So. Polynomials in the variable $\partial_t$ with coefficients in the polynomials in $t$.

OK

@JonasTeuwen Mariano's thesis is about Hochschild Cohomology of algebras of differential operators, if you're interested.

And Ore extensions are like differential polynomial rings.

@PeterTamaroff PhD thesis? Scary stuff.

I don't even know if I'm translating that correctly.

@JonasTeuwen Yes, PhD.

Oh man have to tinkle like so bad! Brb.

Sudden Tinkling Urge.

Mm.

I asked and answered the question "does an infinite dense subset have the same cardinality as the set itself" while peeing!

@JonasTeuwen Can I be your friend?

@JonasTeuwen and the answer was "no" ???

@OldJohn Ah, such a thing is called a free module! (if it has a basis).

@PeterTamaroff No.

@OldJohn Yeah! Take the ultrafilters on the natural numbers. The natural numbers are dense in that.

Bugger!- you are becoming seriously algebraic now - modules?!?!

And that is a sad panda, Sir.

he's too cool for you

:P

@OldJohn Hmm, now my stuff is discrete so it might be useful. Analysis does not handle that well 8-).

@Charlie I have seens pic of Jonas, Ben, Jayesh, Old John. Jasper and you are missing,

@PeterTamaroff Give me your fb id.

@JonasTeuwen I am fascinated by the interplay between algebra and analysis - like the fundamental theorem of algebra needs some analysis in the proof

@Charlie (and met Mariano)

@JayeshBadwaik Done

@OldJohn Algebra is analysis by Atiyah-Singer 8-).

@PeterTamaroff maybe soon you will see...

@OldJohn I liked the pic with the camera you had.

@PeterTamaroff what is a free lance?

@OldJohn I guess my example might be too hard, but you can have the finitely additive two-valued measures on subsets of the natural numbers and this will have larger cardinality than the natural numbers itself which are dense.

@OldJohn Yay!

@PeterTamaroff next to the moss-covered wall?

@JayeshBadwaik When you work independently, AFAIK.

@OldJohn Yeah.

@PeterTamaroff I meant in your case.

You are not a student yet?

@PeterTamaroff my wife took that pic while I was obsessed with taking a pic of the moss @)

@JayeshBadwaik Yes, but the freelance is w.r.t. tennis

Ahh, I did not interpret that correctly. Nice. If we meet, we can rally around a little bit.

The set of ultrafilters on the natural numbers is the three headed monster as the advisor of a friend of mine states 8-).

@PeterTamaroff don't pressure me!

Does not look like pressurizing to me.

@JonasTeuwen HAHA

More like being curious. And so am I! But you give it if you would want to.

@anon Hello, stranger.

hello strangest

@anon Mind seeing an exercise in Spivak?

I don't have time

@anon Awwww OK

@anon No time for talk! Only math math math.

I have to be to school in an hour and then I work 8 hours until 5am and then I have school again. bleh.

no rest for the wicked

@anon ?????

@JayeshBadwaik Don't worry! He is living the American Dream.

@anon Oh! Who will you vote for?

I'm leaning towards either Obama or not voting, haven't thought it through much

@anon I heard Romney wondered why airplanes didn't have windows that could open...

@PeterTamaroff that was a joke.

I couldn't care less about tangential distractions like that.

i almost forgot that i had to vote this sunday ...

@JayeshBadwaik I see now.

@OldJohn So. Perhaps the thing $\mathbb Z[t, \partial_t]$ is actually just $\mathbb Z \oplus \mathbb Z[t] \oplus \mathbb Z[\partial_t] \oplus \mathbb Z[t] \otimes \mathbb Z[\partial_t]$.

Kinda all nice things.

So, that brings me to languages.

Let us take the alphabet $\{t, \partial_t\}$. Then we take all the words on this alphabet. Multiplying words would be just like making compositions of words. Perfect.

So now we would take the free module of that dictionary.

yes - just not sure why you need so many terms in that product

@anon I was thinking that derivatives can't be discontinuous, anon.

@Jay what hour do you wake up everyday?

@OldJohn The first one is redundant I suppose.

I suspect so

Maybe the next two as well. But.

@Charlie Depends, sometimes 8 somtimes 11.

Yeah, here it is.

@OldJohn I was just splitting up my space in nice stuff!

@JayeshBadwaik oh..ok

@JonasTeuwen not a bad idea!

@OldJohn So, actually, this would be on my two letter alphabet the free module with integer coefficients of the words in the dictionary.

@JonasTeuwen I have put the file-system idea here Its not completely well written, and I will update the stuff as it progresses. However, I am thinking, it might be better to TeX it to present it in a clearer idea since it is such a lot of data. So feel free to criticize it as you like.

OK - but tell me again what use this free module can be put to?

@JayeshBadwaik Github supports Markdown.

@OldJohn I am thinking about that!

@JonasTeuwen :))

So a polynomial would be like sentences from our dictionary right?

We would be looking at sentences.

But sometimes we might repeat words to make sure it is well understood - those are the coefficients -

@JonasTeuwen I know, but I tried to get some markdown done, and I was describing tables using simple --- and | but then, it wouldn't accept and newlines have to be specified by two spaces and stuff. It was not working, hence I left. My file is essentially a markdown, but the diagram of a knot does not work probably, so I am not putting is as a markdown.

Not sure what subtracting words is.

Anyway, I am going to sleep now. Good night.

@JonasTeuwen what abut additive inverses of words ...... ?

@JayeshBadwaik Not sure you should have hashing. That is a task of the underlying FS.

@JayeshBadwaik Good night.

goodnight Jayesh

@OldJohn That are negative words right?

yep

I guess that is just like cursing.

@JonasTeuwen Yes, it is a rough draft, i just pushed all my ideas there. lets see how it works out.

@OldJohn gn

So nice and bad words.

@JonasTeuwen ROFL

So a polynomial is a sentence of a stutterer with an anger problem.

@JayeshBadwaik Good night!sweet dreams!

@JonasTeuwen winderful!

@Charlie gn! :-)

and a power series is a never-ending stutter :)

(but they would not exist in your algebra)

@OldJohn Makes good sense right?

@JayeshBadwaik :D

You can consider them, but they are only "formal".

@Jasper :D

You don't care if they converge.

@JasperLoy Jasper, does this make sense?

So basically the stutterer dies.

indeed - all the best power series are just formal :) (avoids nastiness like convergence consideratiuons)

@OldJohn You could also see it as the monkeys trying to write Hamlet.

@JasperLoy Let $f$ be a function defined on $[a,b]$. If there exists $g$ defined on $[a,b]$ such that $g'=f$ for each $x$, then $f$ is continuous.

@OldJohn Those are the power series.

:)

@JasperLoy no,you know this too :P

@PeterTamaroff that can't be true, can it?

@OldJohn I think it is true.

@OldJohn When I say "defined" I mean $f(x)$ exists for each $x\in[a,b]$

@JasperLoy excuse me?

@OldJohn This allows us to rephrase analysis problems concerning power series into a psychiatric problem which can be solved using antipsychotics.

yep - still trying to find a conter-example to that ...

I think the analogy (except my personification) can hold quite well combinatorially speaking.

@JonasTeuwen too deep for me tonight :(

@OldJohn Extreme anger can be "solved" using antipsychotics.

@JasperLoy Yes. Closed intervals.

don't mind if i annoy or disturb you.it means i like you

@JonasTeuwen I can believe that

@PeterTamaroff Yeah. Like you write down what the derivative of $f$ is. Then you multiply by '$h$' and apply the product rule for limits.

What problem do you want to solve?

Oh, right. So you have a function which has a primitive and you want to prove it is continuous?

"Here is an explicit example of a function which is continuous and differentiable, but does NOT have a continuous derivative."

must get some sleep - night all

@OldJohn Night,old!

@Charlie night!

Kinda you make a polynomial times some Heaviside distribution and then you get another function and start messing it up real good by adding perturbations of the first!

@OldJohn Bye!

@JasperLoy Dread. $\sin(1/x)$ always fucks things up.

Or how about the function which is $0$ up to $0$ and then goes up as $x$. Just not differentiable in $0$. Can be fixed.

But you must do it quite... carefully.

@PeterTamaroff Yeah if not add some reciprocals of bad stuff!

Heeeya

Sup Gaylord!

Just chillin being gay as usual.

@JonasTeuwen How about you Scoopface?

@N3buchadnezzar N3!

@Jas Post your question on main...

well.i'm leaving!A good day /night for everyone!

@JasperLoy :P

@N3buchadnezzar Yea.

@OldJohn Oh no! My sentences have to be the same if I permute words. We need a very advanced language where the word order does not matter.

@Charlie G'night

@JonasTeuwen German!

@N3buchadnezzar Smart! Using the wrong order makes you sound retarded in Dutch but comprehensible! The same thing happens in German except that you always sound retarded. Smart thinking!

And you sound angry, so no one wants to argue with you.

Yeah. So in German word order does not matter.

Now we need a language where neither the word order, nor what words you use matter.

@N3buchadnezzar That is easy, you might also want to state what you want to be able to say!

@JasperLoy That is debatable!!!

@JasperLoy What strange ideas are you having?

Okay, then we wont ask

Could someone do me some justice?

Good night guys!

@MarianoSuárez-Alvarez Hola =)