Mathematics - 2012-06-05 (page 1 of 6)

robjohn

00:14

@anon I came up with a different looking answer, but the few cases I've checked seem to be correct.

anilorap

I am working on the understanding of the proof of the sum of four squares theorem by Minkowsky.. i cant understand the matrix...

robjohn

@anon The case that mine handles without canceling infinities is $a=1$ which gives $-\tfrac14\log(2)$

anon

I see.

I was lazy and just blindly calculated :)

robjohn

@anon I will have to see if I can bring mine to your form.

anon

Anyone familiar with group theory? I'm wondering: How does $|(G/H)^U|=|(G/H)^V|$ for each $H\le G$ entail that $U,V$ are conjugate?

Antonio Vargas

00:24

@Jordan, what problem are you having plugging in 1?

Have any other chrome users been having issues with Mathjax these past few weeks? Any TeX won't render after hitting "show comments" or if someone edits an answer while you're reading it. It seems like the first time a page loads is the only time it'll render things.

anon

I think I've been having that at least a little, but I barely notice because I don't really care I guess.

Antonio Vargas

The only time it really trips me up is when expanding comments. There's no way for me to see any TeX below the first few comments which display before you have to click show more.

Chrome and Mathjax haven't been playing nice recently...

Peter Tamaroff

00:51

@AntonioVargas math.SE is crashing everyday I use it.

I'm writing a question and BAM

the lego dead face appears.

Though the page reloads nicely and the draft is saved.

Dylan Moreland

01:11

@anon Everything is finite?

anon

Yes.

leo

hi there!

anon

01:25

@DylanMoreland For reference, this is my adaptation of the first bullet point of "Essential properties of the canonical homomorphisms" pg 2 of Applications of Burnside Rings in Elementary Group Theory (obviously I haven't got very far in!)

Benjamin Lim

@DylanMoreland Hi

I have a quick question, I would like to show that there are no biquadratic extensions of finite fields

say I have $\Bbb{F}_q(\sqrt{a},\sqrt{b})$

Now I want to say somehow that $x^2 - b$ is reducible over $\Bbb{F}_q(\sqrt{a})$

I know that $\Bbb{F}_q(\sqrt{a}) \cong \Bbb{F}_q(\sqrt{b})$

under the isomorphism $\phi $ sending $\sqrt{a}$ to $\sqrt{b}$

Eugene

@BenjaminLim just spitballing here. don't you also have that $\Bbb{F}_q(\sqrt{a}) \cong \Bbb{F}_q[x]/(x^2-a)$?

@Gigili oh i missed this. no i wasn't sulking. i had a seminar to attend and i had to run.

Peter Tamaroff

01:42

@BenjaminLim Benjamin, some chats ago you proposed a problem about $\varphi(n)$ and $\cos \frac{2\pi}{n}$

What was it?

anon

Find $n$ such that the latter quantity is rational.

Peter Tamaroff

@anon I see there is only a finite amount of $n$, right?

Dylan Moreland

@BenjaminLim There's only one extension of $\mathbf F_q$ of a given degree.

anon

@PeterTamaroff Yes. It has to do with cyclotomic integers, which is where $\varphi(n)$ makes its appearance.

Peter Tamaroff

@anon Cyclotomic? Right, since $\cos \frac{ 2 \pi}{n}$ is the real part of the $n$th root of unity.

Dylan Moreland

01:51

@anon Oh, are you looking at the real subfield?

Eugene

@Gigili sorry for the misunderstanding.

anon

I don't know much field theory. Apparently the sketch is: $$[\Bbb Q(\zeta_m):\Bbb Q(\zeta_m+\zeta_m^{-1})]=2\implies \varphi(m)\le2\implies m\in\{1,2,3,4,6\}.$$ But I'm not familiar with the proof.

(On the assumption $\zeta_m+\zeta^{-1}_m$ is rational)

Peter Tamaroff

@anon What is this $$[\Bbb Q(\zeta_m):\Bbb Q(\zeta_m+\zeta_m^{-1})]$$?

I know $\zeta_m$ is the $m$th root of unity.

anon

The index of the subfield (latter) in the cyclotomic field (former), ie the dimension of the former as a vector space over the latter.

Peter Tamaroff

@anon Well, I know nothing about that. Yikes.

leo

02:19

see you

!

Peter Tamaroff

@Eugene I'm really liking Burton's expositions!

@MarianoSuárezAlvarez Please tell me you're around!

Mark Dominus

02:42

I just got an inbox notification of an 8-hour-old comment that I had seen 8 hours ago.

I just got them again. Odd.

Dylan Moreland

The inbox is pretty wonky.

Peter Tamaroff

@MarkDominus Happens to me everytime.

Mark Dominus

This is really nice. Usually with these wacky limit questions you can just eyeball them and guess the answer, but for this one I at least have no idea. math.stackexchange.com/questions/154058/…

Peter Tamaroff

@MarkDominus LOL

Use logs like nukes.

Mark Dominus

I was about to suggest that.

Dylan Moreland

02:49

M-me too.

I want to post an answer here but things with Makoto always seem to get kinda messy.

Peter Tamaroff

@DylanMoreland Why so?

Benjamin Lim

03:05

@DylanMoreland Yes up to isomorphism

@PeterTamaroff Find for which natural numbers $n$ is $\cos(2\pi/n)$ rational

but then how does that mean that $x^2 -b$ splits in $\Bbb{F}_p(\sqrt{a})$?

Dylan Moreland

@BenjaminLim Well, it's slightly better than that. There is a unique extension of degree $n$ inside any algebraic closure, or however you want to put it.

To put it another way, the elements of any quadratic extension are roots of the polynomial $X^{q^2} - X$.

Can't have more than $q^2$ of those! I'll try to think of something more constructive.

Benjamin Lim

yes

That's the confusion I'm trying to iron out now!

@DylanMoreland Isn't the algebraic closure of a finite field a direct limit of something?

Dylan Moreland

Yeah. You don't really have to worry about that here.

Benjamin Lim

right

but somehow

Dylan Moreland

All algebraic closures are direct limits, really. Or at least contained in one.

Benjamin Lim

03:15

Is there any useful information we can get from a perspective like that?

Dylan Moreland

I think of it as being interesting because it's dual to the Galois group side, which involves inverse limits.

But I don't know if it's overly interesting.

Gotta walk the dog, be back in a few.

Eugene

@PeterTamaroff i do prefer burton's exposition of elementary number theory. apostol can be quite unreadable in that aspect

Benjamin Lim

@Eugene yes that is correct what you said

Eugene

@DylanMoreland i was thinking about makoto's question too but stopped myself for the same reason.

@BenjaminLim forget it dylan's way seems easier.

Peter Tamaroff

@BenjaminLim Yes anon reminded me. But it seems far from my knowledge.

Benjamin Lim

03:18

@PeterTamaroff You need to know some field theory to tackle that.

And well yes you also need to know that $[\Bbb{Q}(\zeta_n):\Bbb{Q}] = \varphi(n)$

Peter Tamaroff

@BenjaminLim I don't know what the LHS represents.

Benjamin Lim

ok don't worry then.

That is basically the way to the proof

you need to know field theory like I said.

elementary problem, high-level methods to tackle the proof :D

Peter Tamaroff

@BenjaminLim Is it an undergrad course?

Benjamin Lim

at my university yes

I have a field theory/galois theory final in like 9 days!

Peter Tamaroff

@BenjaminLim OMG OMG OMG =P

How are you feeling?

Benjamin Lim

03:22

not bad

I am quite okay with galois theory

Peter Tamaroff

@BenjaminLim But is Galois and Filed Theory the same/equivalent?

Benjamin Lim

well not really. Galois theory builds on field theory.

Eugene

@BenjaminLim there's a whole thread on problems like that

@PeterTamaroff no

Benjamin Lim

For example consider the problem of asking whether or not $\sqrt[3]{2}$ is contained in any cyclotomic extension.

this is a problem in field theory

@PeterTamaroff by galois theory the answer is like 4 lines !!!!!!

Peter Tamaroff

@BenjaminLim You analyze a polinomial right?

Benjamin Lim

03:25

In galois theory? Kinda

@Eugene Here's the proof. Suppose $\sqrt[3]{2}$ is in $\Bbb{Q}(\zeta_{n})$ for some $n$

then we have $\Bbb{Q}(\zeta_n)/\Bbb{Q}(\sqrt[3]{2})$ being a galois extension with galois group $H$

a subgroup of $\Bbb{Z}/n\Bbb{Z}^{\ast}$

but then this is an abelian group so $H$ is normal

however this means that we can form $G/H$

where $G$ is the galois group of the whole extension

and this means that $\Bbb{Q}(\sqrt[3]{2})$ is a galois extension of $\Bbb{Q}$, contradiction

@PeterTamaroff @Eugene that's the proof

Peter Tamaroff

@BenjaminLim Well, maybe in some time I'll understand it !!

Benjamin Lim

But it's pretty cool eh?

Galois theory reduces the field theory problem to one in a few lines :D

Peter Tamaroff

@BenjaminLim What is $\Bbb Q(x)$ for any $x$?

Benjamin Lim

is $x$ an indeterminate?

Peter Tamaroff

@BenjaminLim I'm asking since you used it twice, for $\zeta_n$ and for $2^{1/3}$

Benjamin Lim

03:33

Oh that's just the smallest field containing $\Bbb{Q}$ and $\zeta_n$ say

in other words all linear combinations of powers of $\zeta_n$

Peter Tamaroff

@BenjaminLim Ok.

So let's see if I can remember.

A field is a generalization of group and ring right?

Like a higher order contstruct.

Benjamin Lim

well it's a generalisation of a ring

in a ring you don't have inverses

Peter Tamaroff

@BenjaminLim Right.

Benjamin Lim

in a field you always do for every non-zero element.

a group has only one operation

a ring two

Peter Tamaroff

@BenjaminLim I.e. say $(+,\Bbb N)$ is a group.

And

Benjamin Lim

03:36

yes

Eugene

@PeterTamaroff that is not a group.

Peter Tamaroff

$(\Bbb R,\cdot,+)$ is a ring?

@Eugene Why not?

Maybe the order is wrong.

Eugene

no additive inverses.

Peter Tamaroff

I should've written

@Eugene Isn't $+$ an operation in $\Bbb N$?

Eugene

firstly, $0 \notin \Bbb{N}$

Peter Tamaroff

03:38

@Eugene Oh, so there is no identity.

Eugene

secondly for every $n \in \Bbb{N}$ there are no inverses.

Benjamin Lim

@PeterTamaroff Another similar problem: How would you determine if $\sqrt{3} \in \Bbb{Q}(\sqrt[3]{3},\zeta_3)$?

Peter Tamaroff

Then I'm confusing group with something else.

Let me check

Benjamin Lim

@PeterTamaroff Sorry I think you meant $\Bbb{Z}$

that is a group under addition

it's also a commutative ring

an integral domain

in fact a principal ideal domain

and hence also a unique factorisation domain too!!

Peter Tamaroff

@BenjaminLim LOL.

You got carried away.

Benjamin Lim

03:40

hahahaha

sorry I was in the middle of determining whether $\sqrt{3}$ was in the field

so I kinda glossed over the fact that I assumed by $\Bbb{N}$ you meant $\Bbb{Z}$?

Peter Tamaroff

For $(S,\star)$ to be a group, I need $(1)$ associativity $(2)$ a neutral element $(3)$ inverse for any $s \in S$.

Benjamin Lim

the key thing in the problem above is that $S_3$ has only one subgroup of order 3 in it :D

yes

inverse under the $\ast$

what was the latex code for your star?

Peter Tamaroff

@BenjaminLim \star haha

Benjamin Lim

ah ok

Peter Tamaroff

Mostly I find out new codes in LaTeX by using simple words.

Benjamin Lim

03:43

@PeterTamaroff Galois theory is very powerful

Peter Tamaroff

@BenjaminLim I am amused by how much Galois accomplished in so little.

Benjamin Lim

well yes

it is very powerful

especially galois correspondence

Peter Tamaroff

Image what'd have happened if he didn't die so young!

Maybe now we'd be much further advanced.

Benjamin Lim

@PeterTamaroff en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory Careful son this is extremely powerful this stuff, highly radioactive thermonuclear weapon, you launch it and you'll alert DEFCON 1

Peter Tamaroff

@BenjaminLim What does that study?

@BenjaminLim With a great power....

I love $\mathcal FT$s!

Benjamin Lim

03:46

huh?

Peter Tamaroff

FUNDAMENTAL THEOREMS

Benjamin Lim

@PeterTamaroff You wanna know something?

Peter Tamaroff

@BenjaminLim Sure.

Benjamin Lim

That there has to be one of the most powerful things I know in my life.

Peter Tamaroff

@BenjaminLim Powerful in what sense?

Benjamin Lim

03:47

when you understand it you will feel the power man

Peter Tamaroff

@BenjaminLim Arrrrrrrg. Can't wait, dude.

@BenjaminLim Was it you that gave a talk some days ago?

Benjamin Lim

yes :D

How do you know? Do you have facebook?

Peter Tamaroff

@BenjaminLim Me? I have a facebook, but you told us! And even made a question about it if memory serves.

Benjamin Lim

let me add you

Peter Tamaroff

@BenjaminLim To Facebook?

Benjamin Lim

03:50

yes

Peter Tamaroff

@BenjaminLim Just a sec. I'm running a search on your police records.

Benjamin Lim

what the hell??????????????????????????????????????????????????????????

Peter Tamaroff

@BenjaminLim LOL JK

Should I send the request?

Benjamin Lim

yes

Peter Tamaroff

@BenjaminLim I can't find you.

Benjamin Lim

03:52

my email

Peter Tamaroff

Oh. Is it in your profile here at SE?

I think I got it

Benjamin Lim

yes

Peter Tamaroff

Wait. maths.anu.edu.au/~isaev ??

Benjamin Lim

no

@PeterTamaroff That's my mentor/supervisor

Peter Tamaroff

@BenjaminLim Oh OK.

Dylan Moreland

03:54

For undergrad? That's different.

Benjamin Lim

@DylanMoreland What are you referring to?

Peter Tamaroff

@BenjaminLim Did you really freak out on this?

Benjamin Lim

no

of course not

Peter Tamaroff

@BenjaminLim I supposed.

Benjamin Lim

@DylanMoreland This is rad: math.uconn.edu/~kconrad/blurbs/galoistheory/splittingisom.pdf

did you add me?

anon

03:55

Since when does Qiaochu do asymptotics? :o

Dylan Moreland

All of KCd's handouts are pretty great.

Benjamin Lim

got it

Peter Tamaroff

@BenjaminLim Oh! You're in Australia?

Benjamin Lim

@PeterTamaroff You can go to the album localisation of bla bla bla

yes I thought you knew that no?

Peter Tamaroff

@BenjaminLim I didn't. All I know is most people here are from the US. And Eugene is from Malasya.

Benjamin Lim

03:57

ah ok

Peter Tamaroff

@BenjaminLim So you're a graduate?

Benjamin Lim

@PeterTamaroff Me a graduate student? You're nuts I'm only first semester second year

Peter Tamaroff

@BenjaminLim Oh haha OK.

Benjamin Lim

why did you think I was graduate student?

Peter Tamaroff

I tend to be impressed by some nomenclature in certain theory.

Benjamin Lim

04:00

hahahahahaha

Peter Tamaroff

And that makes me think it is some higher stuff.

Benjamin Lim

it is powerful

Even now man

Whenever I write down the words $\textrm{Gal}(E/F)$

i feel some vibrations

when I was a young kid I was like what is that???

similarly when I write down a paper "let $X$ be a topological space...."

Peter Tamaroff

@BenjaminLim Ah. I have to learn some point set topology from either Rudin or Apostol or both.

Benjamin Lim

learn from munkres

easy and simply

but mind you that is general topology

Peter Tamaroff

@BenjaminLim Oh, that's another option.

Benjamin Lim

04:02

better to do it in a metric space first

a general topological space has no "distance" in it

Peter Tamaroff

@BenjaminLim Well, I grokke all the metric space constructions. It makes perfect sense. BUt that is PS Topology right?

Benjamin Lim

well I always thought by point set topology they meant general topology done in a general topological space

but that is part of point set topology yes

@PeterTamaroff

3

Q: Diagonal $\Delta = \{x \times x : x \in X \}$ closed in $X \times X$ implies that $X$ is Hausdorff

I think I have solved a problem in Topology by Munkres, but there is a small detail that is bugging me. The problem is stated in this question's title. I will write down the proof and will highlight what is troubling me. We prove by contradiction: Assume $X$ is not Hausdorff. Then there exist po...

general-topology

Peter Tamaroff

@BenjaminLim What I find more interesting is Compactness and Connectedness.

Benjamin Lim

yes

do you know about them?

Peter Tamaroff

I have a small 200 paged book by Bert Mendelson where he covers Metric Spcs $\to$ Topological Spcs $\to$ Connectednes $\to$ Compactness

I have only read the first.

Plus the introductory Set theory (Ch1)

@BenjaminLim I have a nanometrical understanding of connectedness

A vague idea in my head.

Benjamin Lim

04:08

ok

compactness?

Peter Tamaroff

@BenjaminLim There was a blackout. Sorry.

@BenjaminLim I have merely read up to Metric Spaces.

But I have read many times (I think from t.b.) the theorem (which I personally like)

If $X$ is connected and $f$ is continuous, then $f(X)$ is connceted.

Does it go like that?

@MarkDominus What are multisets?

leo

04:26

hi

some random stuff here

Peter Tamaroff

@leo Potato?

leo

@PeterTamaroff chips!

Peter Tamaroff

@leo What are you up to?

leo

I was sending some documments to a mate. Not a big deal. Is it too late in Buenos Aires?

Peter Tamaroff

1:32 am

Mariano Suárez-Alvarez

04:33

the night has just begun

Peter Tamaroff

@MarianoSuárezAlvarez Te estuve buscando estos dias.

Mariano Suárez-Alvarez

Estaba de viaje

Peter Tamaroff

@MarianoSuárezAlvarez Ahora que empecé a escribir en castellano es un poco extraño seguir en inglés!

Mariano Suárez-Alvarez

jaja

leo

jeje

Peter Tamaroff

04:35

@leo Do you get some Spanish?

@MarianoSuárezAlvarez Pudiste ver el mail que te mande?

Mariano Suárez-Alvarez

de cuando?

leo

@PeterTamaroff Claro!

Mariano Suárez-Alvarez

hace unos 5 días que no leo mail.... y estoy tapado de mail ahora :(

Peter Tamaroff

@MarianoSuárezAlvarez Uh, hace un rato ya. El que demuestro las expansiones en series de $\exp$, $\sin$ $\cos$ y $\log$

leo

yo soy de Cuba!

Peter Tamaroff

04:36

Ya te digo la fecha.

Mariano Suárez-Alvarez

sisisi

ese sí

tengo la respuesta escrita en algún lado... debería mandartela

Peter Tamaroff

@MarianoSuárezAlvarez Ah, buenisimo!

Mariano Suárez-Alvarez

últimamente estoy un poco tapado de cosas... perdón por el delay :/

Peter Tamaroff

@MarianoSuárezAlvarez No problem. El de Gaussianos hace mas de un año me debe algo!

Mariano Suárez-Alvarez

:)

Peter Tamaroff

04:38

Me pidio que le mande algo sobre la formula de Viete

Y nada...

Mariano Suárez-Alvarez

andás por exactas?

Peter Tamaroff

@MarianoSuárezAlvarez En el CBC de Martinez.

leo

@MarianoSuárezAlvarez que quieres decir con "tapado de cosas" es como "lleno de cosas"

?

Mariano Suárez-Alvarez

@leo: sí :) demasiados cosas que hacer!

Peter Tamaroff

@leo Como, muchas tareas pendientes.

Mariano Suárez-Alvarez

04:39

@PeterTamaroff, ah

leo

a ya

Mariano Suárez-Alvarez

si alguna vez andás por Exactas, pasá a saludar!

Peter Tamaroff

@MarianoSuárezAlvarez Totalmente! Hoy me devolvieron el primer parcial de Analisis

Va no hoy

Mariano Suárez-Alvarez

y? :)

Peter Tamaroff

Me lo habian devuelto y tenia mal corregido algo.

De 9 a 10 jajaja.

Pero merecidos.

leo

04:40

que bueno! felicidades

Mariano Suárez-Alvarez

bien :)

Peter Tamaroff

@MarianoSuárezAlvarez Muchos $0$ hubo!

leo

@PeterTamaroff y que entraba?

Mariano Suárez-Alvarez

jajaja

no me sorprende :)

Peter Tamaroff

@leo Secuencias, limites, continuidad, derivadas, bolzano, teoremas de valores intermedios, optimizacion. Nada muy complejo.

@MarianoSuárezAlvarez Siempre es asi?

Mariano Suárez-Alvarez

04:42

sip, lamentablemente

Peter Tamaroff

@MarianoSuárezAlvarez Buen, por lo menos no andan dudando los pibes.

Es un no rotundo. O a estudiar, o a elegir otra carrera.

leo

son bonitos esos examenes. Da gusto estudiar eso. Resolver problemas de eso!

Mariano Suárez-Alvarez

jaja

Peter Tamaroff

@leo Si, pero son mas interesantes los problemas menos tipicos

Ya todo se vuelve muy mundano despues de un rato.

@MarianoSuárezAlvarez Vos que enseñas ahi en el pab 1?

Mariano Suárez-Alvarez

ahora, geometría diferencial

todos los cuatrimestres cambiamos de materia

Peter Tamaroff

04:44

Ahh, OK.

leo

@PeterTamaroff si claro. Pero por lo menos la primera vez que uno ve todo eso da gusto. Y cuantos problemas teniae el examen?

Peter Tamaroff

@leo Eran 4. Cuando me lo devuelvan te lo paso, a ver como te sale =)

leo

:-)

Peter Tamaroff

@MarianoSuárezAlvarez Como se manejan con los promedios? De los que vienen del CBC?

Me dijeron que un buen promedio te asegura una buena cátedra.

Mariano Suárez-Alvarez

Not really. Al menos en Matemática vas a la cátedra que te da la gana

leo

04:46

Si yo tuviera que hacer examenes pondría unos cuatro problemas para evaluar lo que se debe evaluar un problema extra de puntos adicionales pero que sea bien dificil

Peter Tamaroff

@leo El extra es a eleccion?

@MarianoSuárezAlvarez A OK. Pero claramente una mejor catedra es mas exigente, no?

leo

no dificil tedioso, si no mas bien dificil bonito :-)

@PeterTamaroff si si

Mariano Suárez-Alvarez

En matemática, salvo por el primer par de materias, todas las materia tienen exactamente una :)

en las que tienen varias cátedras (análisis I, álgebra I) se hace un gran esfuerzo por coordinar

así que mucha diferencia no hay

aunque, claro, depende del docente

Peter Tamaroff

@MarianoSuárezAlvarez Para. Pero entonces no tenemos Analisis II el año que viene no? Los que venimos del CBC.

leo

@MarianoSuárezAlvarez ja ja aquí es igual

Peter Tamaroff

04:49

Tenemos Analisis I.

Mariano Suárez-Alvarez

todos vienen del CBC :)

Peter Tamaroff

Yo creía que el analisis del CBC suplanto lo que era Analisis I.

Mariano Suárez-Alvarez

no

no es lo mismo, de todas formas

Peter Tamaroff

@MarianoSuárezAlvarez Jaja claro

@MarianoSuárezAlvarez Digamos que el Analisis del CBC es mas "Calculo" y Analisis I es mas "Analisis" o lo Apostol/Rudin?

Mariano Suárez-Alvarez

not really :)

el contenido de las materias iniciales de análisis no es mi favorito, digamos

Peter Tamaroff

04:51

@MarianoSuárezAlvarez Uh! Que cubre¿

Mariano Suárez-Alvarez

hace algunos años hubo un cambio en los contenidos de anal. I y II

desde entonces nunca di esas materias

Peter Tamaroff

@MarianoSuárezAlvarez Te fuiste por eso?

Mariano Suárez-Alvarez

si me tocara darlas, las daría—pero no estoy particularmente de acuerdo con lo que se hizo

Peter Tamaroff

@MarianoSuárezAlvarez Ah. Las retocaron para mal?

Mariano Suárez-Alvarez

bueno, mucha gente piensa que para bien: por eso lo hicieron

:P

pero yo hubiera preferido un cambio distinto

Peter Tamaroff

04:54

@MarianoSuárezAlvarez ja claro =D Pero, que cambiaron?

leo

bueno, pasen buenas noches!

Peter Tamaroff

Tengo entendido que antes el CBC era mas riguroso. Analisis era mas riguroso en si mismo...

Y fueron distendiendolo

Mariano Suárez-Alvarez

inicialmente el cambio se suponía que era empezar en anal. I con cálculo enb varias variables, asumiendo que en el CBC ya habían aprendido cálculo en una variable

Peter Tamaroff

@MarianoSuárezAlvarez OK.

Mariano Suárez-Alvarez

ahora, honestamente, no sé bien qué es lo que hacen

Peter Tamaroff

04:56

@MarianoSuárezAlvarez Hm. Ya me enteraré.

Me decepcionó un poco como introdujeron la integral el otro dia! =P

Un amigo que cursa en otra comision me mostró como se la explicarion a ellos, y era basicamente la explicacion de Spivak. Que no esta nada mal.

Mariano Suárez-Alvarez

:)

Peter Tamaroff

Pero a nosotros nos mandaron una cosa medio rara.

Mariano Suárez-Alvarez

vás y la leer del libro de spivak y listo

Peter Tamaroff

Si, yo ya lo habia leido y releido muchas veces el Spivak.

Le debo mucho a ese libro.

Es el que mas use. Ahora estuve repasando un poco lo que es la Completitud de $\Bbb R$, las cotas superiores, viendo un poco si incorporo lo que es el $\limsup$/$\liminf$ que esta ahi

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$Mathematics$ Mathematics