Mathematics

@Analysis Make your question more apparent to the skimming reader

state directly what you want and then highlight it with a quote box (using >)

That is a good idea. I feel like that means I shouldn't ask multi questions then

i keep coming to the end of your question and not really knowing what you were asking. It might be apparent if I went back and read carefully but most people will not take the time unless they are interested.

@Analysis it sure does

@Alexander I assume you know the 'double' jump result in random graphs, right?

@Studentmath I don't.

12:30 AM

That in $G(n,p)$ when $p=(1+\epsilon)/n$ there's a unique component of linear size to n, and when $p=(1-\epsilon)/n$ all components are of logarithmic size?

@AlexanderGruber Is this better " math.stackexchange.com/questions/939555/… "

@Studentmath Oh, yeah. I have seen that.

@Analysis That's a good summary - now the best thing would be to go halfway between before that, and now, by adding your own question as an example

I've been wondering why everyone focus on that, whereas Erdos in his original paper didn't find it interesting enough to focus on, and, working in G(n,m), he searched for the specific probablity of the largest component having $n-k$ vertices in it. Just felt like sharing this, no real question here..

"What do I do when -L<x<0?" then below that "for example, if I wanted to compute f(x)=2x^2..."

Then everyone taking his paper focused on that instead of what he thought was way more interesting (and includes in a way that result, with a bit of work)

12:33 AM

that way people know right away what the point of the question is, and they also have a specific example to work with. Makes the answer easy to write (for those who know how to do it).

@AlexanderGruber That also makes sense. Would I put the example in '>'?

@Studentmath People probably just think it's weird. It's unusual to see that kind of thing.

Also maybe it is more accessible than the rest of Erdos's thoughts in that paper

@Analysis I'd leave it out of > for this one, places more emphasis on your summarizing question

Yes. I guess accesibility plays a great part in it, too.

I can't bring myself to sit down and truly understand his proofs there just yet, it feels a bit of a burden.

But probably because the paper I have is a copy of the 1959 paper, the letters themselves make my eyes burn

@Analysis for examples of how to write like this you may want to check out some of my questions, i use this sort of pattern in nearly all of them

(though, sometimes i ask questions without showing any effort and most people can't get away with that)

@Studentmath I really dislike most of Erdos's proofs.

that guy thinks much differently than I do, I find his process hard to understand

(not that they're not good proofs, I'm just a different type of thinker)

Precisely - I find his process hard to understand.

Yeah. I read a very nice proof of the double-jump result, it wasn't much simpler, it involved some rather long-to-prove lemmas and so on. But it fit my line of thinking, so I understood where they were heading the moment they wrote the theorem itself.

12:41 AM

yeah :) sometimes the challenge is more writing it out than understanding how it will be done

those are always good papers

Thank you for advice Alexander

@Analysis No problem, hope you get some answers. I'd help you out if I knew much analysis.

Indeed.. eitherway, glad I went with this subject. It's really interesting.

@AlexanderGruber!

@Studentmath...!

@Balarka !

12:51 AM

Ah good you're here =)

Soon hitting the bed (it's 4 am here) but yeah :P

It's 6:30 here =P =P

In the...?

Morning?

Yes. AM.

Really? Only 3 hours ahead?

I would expect more.

Are you in Western India?

12:52 AM

I do believe it's more than that

@Studentmath What's the date in there?

21/9

03:55

WAT

So we are 3 hours ahead...!

It makes sense if you are not in Eastern India

Erm.. I forgot what direction is East.

I think Ukraine is one hour ahead, so yeah. 3 would fit.

East is China

West is US

12:54 AM

I dunno china. Is it $\infty + i\infty$ or $-\infty - i\infty$?

If you have a normal map at home and you compare India to these places.

Or the conjugates?

That would be Australia.

So what is East? Be explicit.

I think I know where Iran is. If you take up the coordinate origin to where I am, then it's on $-\infty + i\infty$

Okay, look at a map of India. draw an imaginary line from the long bottom of it all the way up. If you live right to that place, you are in eastern India. Otherwise, by the logic that every statement is either true or false, you are in western.

12:57 AM

@Studentmath Erm. What is right again?

The hand without the watch.

I don't wear watch.

The hand you write with.

Is it $\Bbb R$ or $- \Bbb R$?

@Studentmath Ah good enough

@BalarkaSen north pole's up and south pole's down

12:58 AM

OK, OK, got that.

I am on the right then @Studentmath

thus eastern india is right because we use the right hand rule

"Eastern India"

I forgot why we started this.

LOL, @Studentmath

The 3 hours difference, I think

Yeah well,

Wait, it's 7:00 now, right?

1:00 AM

6:30. Close enough.

No no.

You can't live between 2 time-spaces.

What is this place.

Erm. Kolkata/Calcutta I think.

In West Bengal.

Well move it 30 minutes forward or 30 minutes backwards.

We are 3 hours and 30 minutes apart

But 6:30 - 4:00 is NOT 3:30.

2:30. I am tired.

1:03 AM

Peace out, @Studentmath

You are UTC+05:30. I never thought you could be something less than full hour away.

Yeah, good night all!

@Alexander Whoa. You changed that illuminati triangle.

@BalarkaSen I didn't.

It must have been the Illuminati.

Oh. Might be a trick of eye then.

Alizter

I need to have better intuition than $$\sum_{n=1}^\infty \frac{\sin(\sqrt n)}{n^{3/2}}\approx1+J_2(\pi)$$

time for sleep

1:14 AM

Whoa @Alizter

What's the context?

@Alizter here

$$\sum_{n=1}^\infty \frac{\sin(\sqrt n)}{n^{3/2}}\approx1+\mathfrak{J}_2(\pi)$$

does that help?

ahahha

Hmm. $\mathfrak{J}$ looks much like $\Im$

plans for a notation abuse

Any problem to think about @AlexanderGruber? And when I mean problem, I mean ring theory/field theory/galois theory.

@BalarkaSen Let $\mathcal{J}\subset \Im \subset \mathfrak{J}$ be finite extensions, and consider elements $\mathit{j},\mathcal{j},\mathfrak{j}$...

$\mathfrak{j}$ looks familiar...

Hmm. Dagger!

anon

1:36 AM

@Sush if the discriminant is nonnegative and the middle term in the quadratic polynomial (in y^2) is nonnegative, then for all x there exists a y for which (x,y) is on the curve. that can't be bounded.

@BalarkaSen i don't have a problem but i do have some sets.

Let $$\mathcal{C}_{n}=\{z\in \mathbb{C}:p(z)=0\text{ for some polynomial }p\text{ whose coefficients are }n^\text{th}\text{ roots of unity}\}$$

(so, for clarity: we'd have $\mathcal{C_n}=\bigcup_{m=1}^\infty C_{n,m}$, where $C_{n,m}=\{z\in \mathbb{C}:\exists n^\text{th}\text{ roots of unity }a_1,\ldots, a_m\text{ such that } a_0+a_1z+\cdots +a_nz^m=0\}$)

usually I'll be talking about $C_p$ for $p$ prime, and also it seems to help to view these as points in $\mathbb{R}^2$

so, here's my question to you. How can we Galois theory these?

certainly there is something to talk about.

for example, we could look at the splitting field.

i posted some pictures of $\mathcal{C}_{n,M}=\bigcup_{m=1}^MC_{p,m}$ on here before. They seem like interesting sets.

here's $\mathcal{C}_{3,8}$, for example. What can you tell me about $\mathcal{C}_3$?

(that's also a question you might be interested, @anon)

Semiclassical

2:29 AM

@IanMateus: Just noticed your +100 bounty on Olivier's log-integral problem. Hope you have more response with it than my +50 bounty on the same...

HI

I have a little question, suppose G acts on X, a have checked an old exam and it define $X_G=\{x: gx=x for all x\} someone knows what is the name of that. X does not have structure of group necessary is that set well defined? what characteristica can have?

Well more specific. If we let $G$ a p- group and X a G set, show that $o(X) = o(X_G) mod p$ where X_G is defined as above.

o is order.

I'm a little confused about the set X_G, maybe is a typo

Someone?

2:47 AM

I wouldn't know sorry, and others are seemingly inactive

I might know, but my latex isn't rendering.

the definition of the set $X_G= \{x\in X: gx=x for all g \in G\}$

From what I can tell you are talking about group stabilisers

Or you are talking about orbits, I can

't tell since I can't read unrendered latex well

Not is not, is like a stabilizer but over the set, I don't know if is clear jaaaja but X doesn't necessary have structure of a group

I had never seen that set before maybe is a typo

The set is in word: the set of all x in the G set s.t., g of x is x for all g in the group acting on X

but the problem didn't mention that X has a group structure or something else.

What is $X$ here?

an element in the X which is a G set

2:52 AM

$X=X_G$

The entire problem is something more complicated but I'm a bit confused with the definition of X_G

@JoseAntonio What's the problem?

G acts on X and X_G is the set of all the elements in X s.t. are unchanged under the action of G. It´s like a stabilizer but the set lies in X which not necessarily is a group. That's confuse me

Also, that's usually denoted by $X^G$.

@robjohn Could you answer my new question, related to my question you commented on last night: math.stackexchange.com/questions/939555/…

2:55 AM

@JoseAntonio Why does that confuse you? $X^G$ is the set of fixed points of the action.

The entire problem says Suppose X is a G set and G is a p group shows that $o(X) = o(X_G) mod p$ where $X_G=\{x\in X: gx=x for all g\in G\}$

@JoseAntonio Yes.

Do you know that $X$ is partitioned into orbits under the action of $G$?

Yes

That is $X=\bigsqcup \mathcal O(x)$.

This means that $|X|=\sum |\mathcal O(x)|$.

my little o, is not the orbit is of order

2:58 AM

Now, when is it true that $\mathcal O(x)=\{x\}$; i.e. the only element in the orbit of $x$ is $x$ itself?

This has to do with $X^G$.

@JoseAntonio I know.

$\mathcal O(x)=\{x\}$ means $gx=x$ for every $g\in G$. That is, $\mathcal O(x)=\{x\}\iff x\in X^G$. So $|X|=|X^G|+\sum \mathcal O(x)$ where the sum runs through nonsingleton orbits.

So far so good?

$X^G$ is what I name $X_G$, right?

Yes.

So we can write $X$ has the sum of all fixed points (= size 1 orbits) and orbits of size >1.

$O(x)=\{x\}$ only when the group action doesn't change the element, hence it is a fixed point.

Now, you know that $|\mathcal O(x)|=|G:{\rm stab}\; x|$.

In particular, $|\mathcal O(x)|$ divides $|G|$. Yes, @JoseAntonio ?

In fact $|G|/|\mathcal O(x)|=|{\rm stab\;} x|$.

@robjohn What textbook would you recommend for learning Laplace and Fourier?

3:02 AM

Here we're assuming $X$ and $G$ are both finite.

@JoseAntonio

Ok I think I got it

This means $|\mathcal O(x)|=0\mod p$. Reducing the equation we got above, we obtain $|X|=|X^G|\mod p$ for $G$ a $p$-group.

Bye bye!

That's set has a particular name?

bye thanks for your help

@PedroTamaroff Thanks for your explanation.

Hello all.

Hello @Awerth

3:11 AM

Hello @Analysis.

@AlexanderGruber, are you around?

what text recommend for algebra? I love Artin's book, but doesn't have important topics as the semidirect product

What is your background?

@AWertheim Somewhat.

@JoseAntonio Dummit and Foote is the only way to live.

Dummit and Foote is very very good, but a bit dry.

It also defers almost all discussions of category theory until the end.

@AWertheim You're not supposed to be drinking it.

3:14 AM

@AlexanderGruber That explains my difficulties with it ;)

I thought they did some categories along with the commutative algebra in the middle

am i remembering that wrong?

I am actually going through Dummit and Foote rather extensively at the moment, funny enough. It is hard to beat in a lot of ways.

Mmm, I think the commutative algebra comes more at the end. Chapters 15-19, I think.

I really love Artin's book but it's a pity that his book doesn't have all the topics. I think I will use Dummit and Foot

@JoseAntonio what do you want to know about semidirect products?

By then, you've seen groups, rings, modules and fields with nary a discussion of categories or functors.

3:16 AM

@AWertheim that seems like a good way of doing things to me