Mathematics - 2013-04-02 (page 1 of 3)

I would argue that anything containing Z/k for some unbounded set of positive reals k must be everywhere dense, as d(x,Z/k)->0 as k->infinity for any real x.

Peter Tamaroff

@MarianoSuárez-Alvarez Let me see...

Mariano Suárez-Alvarez

2:47 AM

no, that is not it Peter

Peter Tamaroff

@MarianoSuárez-Alvarez Oh, sorry. You wrote $g\in \Bbb Z$, I got confused...

Mariano Suárez-Alvarez

@Expert, that is precisely what I am arguing... :-)

Expert

indeed

Peter Tamaroff

@MarianoSuárez-Alvarez I'm looking at it.

Mariano Suárez-Alvarez

no rush :-)

Peter Tamaroff

2:49 AM

@MarianoSuárez-Alvarez OK, I think I got it.

$g\lfloor x/g\rfloor$

Mariano Suárez-Alvarez

well, an integer cannot be the answer to my question :-)

Peter Tamaroff

That is $0<x-g\lfloor x/g\rfloor < g<\epsilon$

Mariano Suárez-Alvarez

but, up to that, yeah

:-)

Peter Tamaroff

@MarianoSuárez-Alvarez I missed the $g$, my bad.

I mean, I wrote it on the paper.

Forgot to add it here =)

Great.

Mariano Suárez-Alvarez

no, I am saying that a number cannot answer my question

Peter Tamaroff

2:51 AM

@MarianoSuárez-Alvarez What was your question?

Mariano Suárez-Alvarez

which was «can you see why....?»

anorton

@Expert Who are you, really?

:D

Expert

it's not late enough for drinking and existentialism

Peter Tamaroff

@MarianoSuárez-Alvarez Well, I can see why because I provided one, but... what is your point? =/

Mariano Suárez-Alvarez

I always find it strange when people answer questions in ways which are incompatible with the question :-)
Like «Do you want a or b?» answered by «Yes!»

Peter Tamaroff

2:53 AM

@MarianoSuárez-Alvarez HAHAHAHAH

anorton

@Expert ah ha! I figured it out by looking at comments...

Mariano Suárez-Alvarez

soo now you can do that for all $\epsilon$s and tha proves density.

Peter Tamaroff

@MarianoSuárez-Alvarez Great.

Expert

@anorton or you could have noted my name is in italics, and that I am not robjohn (this year, anyway).

Peter Tamaroff

So every additive subgroup of $\Bbb R$ is either discrete or dense.

@MarianoSuárez-Alvarez Thank you.

anorton

2:54 AM

@Expert ooh... I didn't know why certain users (namely, two of them) had italic usernames here...

Mariano Suárez-Alvarez

yup

anorton

until now

Mariano Suárez-Alvarez

another way of saying that is that the proper closed subgroups of $R$ are discrete.

Peter Tamaroff

@MarianoSuárez-Alvarez Right.

@MarianoSuárez-Alvarez Have you ever heard of Truesdell's Essay?

Mariano Suárez-Alvarez

not really

Peter Tamaroff

2:57 AM

@MarianoSuárez-Alvarez It is called "An essay towards a unified theory of Special Functions."

Mariano Suárez-Alvarez

nope

ah, actually yes

i've see the book

Peter Tamaroff

@MarianoSuárez-Alvarez Well, I am interested in its contents. Bill Dubuque told me it is "essetially Lie Theoretic" and I am interested in the connection of Algebra with the theory of Special Functions now.

It is all about "the F equation"

$$\frac{\partial}{\partial z}F(z,\alpha)=F(z,\alpha +1)$$

@MarianoSuárez-Alvarez Congrats on +1000 answers, BTW!

Mariano Suárez-Alvarez

it is Lie theoretic becausse most special functions appear as characters of representations of Lie groups

and/or variants of that

@PeterTamaroff ah really?

Peter Tamaroff

@MarianoSuárez-Alvarez OK, characters as in $f:G\to\Bbb Z$ where $G$ is a group? I have heard of characters of abelian groups so far.

Mariano Suárez-Alvarez

no, not as that

as (restrictions to appropriate subgroups) of trace functions

Peter Tamaroff

3:02 AM

Hmmm. I have no idea what trace functions are. Let me google that...

Mariano Suárez-Alvarez

if $f:G\to GL(V)$ is a group hom., its character is the fuunction $g\in G\mapsto \operatorname{tr}f(g)$.

in usseful situations, the morphism $f$ is sort of completely determined by $f$

Peter Tamaroff

@MarianoSuárez-Alvarez Where $\operatorname{tr}$ is the usual trace of matrices?

Mariano Suárez-Alvarez

and the character is a rather special function, which is determined by its restrictions to appropiate subgroups of $G$

Peter Tamaroff

@MarianoSuárez-Alvarez OK.

Mariano Suárez-Alvarez

when one restricts sufficiently well, then one is leeft with lots of classical special functions

more elaborate special functions (like basic ones and so one) show up inthis way also, but replaacing groups by more complicated objects.

Peter Tamaroff

3:05 AM

@MarianoSuárez-Alvarez OK. I guess "Lie Algebras" are like faaaaaaaaar away in Algebra III? Or are they part of a special course altogether?

Mariano Suárez-Alvarez

they are not covered in the algebra courses

Peter Tamaroff

@MarianoSuárez-Alvarez I went to the library above the buffet the other day... people kept talking like it was a bar. It drove me insane...!

@MarianoSuárez-Alvarez Oh, OK.

Mariano Suárez-Alvarez

you can take "lie algebras and lie groups" as an optative

Peter Tamaroff

@MarianoSuárez-Alvarez Oh, good.

@MarianoSuárez-Alvarez I couldn't solve an exercise from your book.

$$\alpha=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$$
$$\beta=\begin{pmatrix}0&1\\-1&-1\end{pmatrix}$$

You ask to find $\langle \alpha,\beta\rangle $

Mariano Suárez-Alvarez

Ah

by the way

this year we are going to have the book reprinted

Appu

3:10 AM

Hello Mathematics!

Mariano Suárez-Alvarez

and we want to fix bugs in it

Appu

This is the first time for me in StackExchange chat rooms.

Peter Tamaroff

@MarianoSuárez-Alvarez Oh, I can totally help in finding typos and stuff.

Mariano Suárez-Alvarez

so anything, anything, ANYTHING you find in it that needs to be fixed please oh please let me know

Peter Tamaroff

@MarianoSuárez-Alvarez Sure!

@MarianoSuárez-Alvarez IIRC, I found "tranformacion" instead of "transformacion" a lot.

@MarianoSuárez-Alvarez Will you make it into a nice book I can get my hands on?

Mariano Suárez-Alvarez

3:12 AM

yup, the idea is to print more copies than last time

the book is actually used in several places

and they all want to buy it

Peter Tamaroff

@MarianoSuárez-Alvarez Wow. Congrats.

Expert

@Appu welcome

Appu

@Expert Thank you. I got Talk to Expert Dialog many a time yesterday. Is that you who chats?

Peter Tamaroff

@MarianoSuárez-Alvarez May I ask what it was you corrected in the question that made that guy so angry?

Expert

@Appu no, I changed my name/gravatar to match the bot in the spirit of april 1st, gambling on the grace of our esteemed moderators to whisk me back afterwards. (nudge nudge)

Peter Tamaroff

3:19 AM

@Expert hehehhe

Mariano Suárez-Alvarez

let $A=(a,b;c,d)$ be an invertible matrix of integers. $\alpha A$ swaps the rows (and messes a bit with the signs) and $\alpha\beta A$ adds the second row to the first (and changes signs).
Now the numbers $a$ and $c$ are coprime. Let us play the Euclidean algorithm on the ordered pair (a,c).

that is, if $c>a$, then let us substract $a$ from $c$, so get $(a,c-a)$; if $a$ is larger, do the opposite.

Since the two are coprime, we know we end up in $(0,1)$. (I am being very sloppy with the signs...)

This means that multiplying on the left by $\alpha$ and by $\alpha\beta$ the matrix $A$ we know we can get to a matrix whose first column if $(0,1)$ in some order and with some signs.

Peter Tamaroff

@MarianoSuárez-Alvarez OK. I'll read that in detail.

Will Jagy

@MarianoSuárez-Alvarez, I cannot find the immature teenager comment.

Mariano Suárez-Alvarez

-16

Q: A case of petulance at my expense.

I want to share a story with the community, one that I think is showcases some unprofessional petulance. I do so, mostly curious as to how my peers here see it: do they see me as the instigator or victim of this breach of etiquette? Consider this a question of conduct, and not entirely as a mediu...

discussion comment etiquette flagging down-votes

Peter Tamaroff

@MarianoSuárez-Alvarez About what you told me above (revising the book) look at p17, exercise 4.

Appu

3:27 AM

@Expert I could guess it lol.. Because when I chated yesterday I found that it's automated.

Amzoti

Hi All - does anyone know how stack exchange is paid for - the servers, bandwidth, maintenance, programmers ...?

Mariano Suárez-Alvarez

@PeterTamaroff, can you make a list of these and send it by email? Otherwise things get lost :(

Amzoti

What a great concept - but wondering how it is supported

Expert

@Amzoti ads on SO I think, mainly, see http://meta.math.stackexchange.com/questions/6584/where-does-the-funding-for-mathematics-stack-exchange-come-from

Peter Tamaroff

@MarianoSuárez-Alvarez I'm using Adobe to highlight them. I can send the list when I'm done.

Amzoti

3:29 AM

@Expert: Ah - I see - fortunately there aren't many! thx

Appu

gotta work. cya later all.

Mariano Suárez-Alvarez

@WillJagy He called me «esteemed collegue» at some point!

@PeterTamaroff, the outcome is that the two matrices generate $SL(2,\mathbb Z)$.

Peter Tamaroff

@MarianoSuárez-Alvarez I see. I noted it was a subgroup of ${\rm SL}(2,\Bbb Z)$, yes. But that is kinda obvious...

Mariano Suárez-Alvarez

The two belong to it, and iff $A$ is in thatgroup, multiplying it on the left by $\alpha$ and by $\alpha\beta$ mimicking the Euclidean algorithm you can simplify it

It is probably better to notice that $\gamma=\alpha^3\beta$ is $(1,1;0,1)$, which makes for the euclidean algorithm to go much directly.

Will Jagy

@MarianoSuárez-Alvarez, original deleted math.stackexchange.com/questions/347807/… Evidently some comment about pedantry was deleted.

Mariano Suárez-Alvarez

3:35 AM

yup. My petty tyrancy was brought to fore at some point, too.

H. Makholm uses the word modicum, too, by the way :-)

Will Jagy

Modicum. Cool. He's a big shot undergraduate where he is. On the other hand, he is the one asking the questions. I've always thought that should be a factor in choice of language.

Mariano Suárez-Alvarez

sure

but $\{x:p(x)\}$ has a well established meaning :-)

I actually spent a while trying to see what he was asking before noticing that the notation was off

for the 2nd part, I still do notknow what he meant to ask

Will Jagy

OK. I did not really try to read the question. I remember that guy on MO at the Harvard lab that does machine learning, he thought all mathematicians knew what SVM meant.

Then he got really pissed off when we closed his question.

Peter Tamaroff

@MarianoSuárez-Alvarez Que relacion tiene un "Carcaj" con un "Grafo orientado"?

Parecen similares.

Mariano Suárez-Alvarez

@PeterTamaroff, they are the same thing

Peter Tamaroff

3:47 AM

@MarianoSuárez-Alvarez Que nombre raro! =)

Mariano Suárez-Alvarez

using one or the other only shows a difference of intent

carcaj means exactly the same as quiver in english

or carquois in french

it is the thing where you put your arrows

Peter Tamaroff

@MarianoSuárez-Alvarez Oh.

Mariano Suárez-Alvarez

the term was introduced (in French) by Pierre Gabriel

who loooooved to introduce new terms

Will Jagy

of Genesis?

Mariano Suárez-Alvarez

most of his inventions never caught

but quiver did

Peter Tamaroff

3:49 AM

@MarianoSuárez-Alvarez Sounds nicer in either language but Spanish =(

Mariano Suárez-Alvarez

in portuguese it is also funny

Peter Tamaroff

@MarianoSuárez-Alvarez "Aljaba" is a nice synonym. I never like the soud "j" makes though.

Mariano Suárez-Alvarez

but if you say aljaba no one will know what you mean :-)

Peter Tamaroff

Like in "jarro", "jugo"

Mariano Suárez-Alvarez

carcaj is probably the only word in spanish ending in j

Will Jagy

3:52 AM

@MarianoSuárez-Alvarez, also Macarenaj

Peter Tamaroff

@MarianoSuárez-Alvarez HAHAHAHA yes.

Will Jagy

The Following is on in a minute

Peter Tamaroff

@MarianoSuárez-Alvarez Oh, wait. Nevermind.

Expert

@PeterTamaroff I recall you making an argument that every finite integral domain was a division ring, but what about commutativity?

Peter Tamaroff

@Expert Yes, yes. I misread!

Or "misrecalled". Whatever.

Expert

3:56 AM

the proof of that involves centers, cyclotomy and counting iirc

Peter Tamaroff

@Expert This book has a proof using Cyclotomic polynomials, Mobius' function, division sums, and cardinality of certain subsets, yes.

Mariano Suárez-Alvarez

there is a discussion of alterantive proofs in the comments to mathoverflow.net/questions/42512/…

the alternatives use more elaborate things, like Brauer groups or the Noether-Skolem theorem.

using N-S is best, IMO

Peter Tamaroff

@MarianoSuárez-Alvarez Is it raining frogs over there too?

Mariano Suárez-Alvarez

yeah

Peter Tamaroff

4:18 AM

@MarianoSuárez-Alvarez I'm guessing you know this one

Expert

Hmm, the early 80s were 30 years ago...

Mariano Suárez-Alvarez

4:38 AM

@PeterTamaroff there you go :-)

Peter Tamaroff

@MarianoSuárez-Alvarez Upvoted.

Rainy nights are not made for sleep!

Mariano Suárez-Alvarez

4:53 AM

Hangin' around, nothing to do but frown;
Rainy days and Mondays always get me down

Peter Tamaroff

►

@robjohn Hey.

How did April Fools go?

Expert

speaking of which. name change plz kthnx

Peter Tamaroff

@Expert plzkthnkxbyes

robjohn

5:09 AM

@PeterTamaroff I don't think I was fooled.

Peter Tamaroff

@robjohn Good.

@robjohn Did you fool anyone?

robjohn

6:38 AM

@PeterTamaroff I didn't really try.

Mohsen Afshin

Can we say $1+\o(1) < 2$ ?

?!?

!?!

what's o(1)?

7:38 AM

o(1) is a langrange symbol

sry landau

Dominic Michaelis

8:24 AM

what is $H^2$ ?

Jayesh Badwaik

8:47 AM

Now that April Fool's is over, the pinned message can be removed methinks. :-)

Dominic Michaelis

8:58 AM

@robjohn isn't your solution essentially the same ? i mean don't you use the same substitution ?

Transcript for

$Mathematics$ Mathematics