Mathematics

The Artist

6:00 PM

@UserX , place it on top of another. And no

Balarka Sen

@UserX Order by numerator/denominator.

Sawarnik

@TheArtist This is getting annoying.

Ice Boy

3 mins ago, by Ice Boy

trolling = teasing

The Artist

@Sawarnik believe me il be a good husband to you...and our horoscope match

UserX

@BalarkaSen elaborate on the Cantor and the numerator/denominator part a little more?

Sawarnik

6:00 PM

I will report to mods if you continue.

Even if you are troll, or not.

Most probably you are though. Not Anastasiya though.

Ice Boy

3 mins ago, by Ice Boy

@Sawarnik put him on ignore

The Artist

@Sawarnik are you serious? :o learn to be more patient :) il stop

@Sawarnik I stopped :)

Sawarnik

@IceBoy Yeah, I ll just do.

robjohn

@RandomVariable why are you breaking up the numerator? you need to use the fact that the denominator is a factor of the numerator, or you do need to handle the singularity

Sawarnik

I thought that he/she would stop after sometime.

Balarka Sen

6:02 PM

@UserX List all the rationals (with increasing numerator) with denom and numerator that add up to 2, then 3, then 4, etc.

The Artist

@Sawarnik i said I stopped :)

Balarka Sen

And cancel out multiplicities.

Ice Boy

@Sawarnik save yourself

Sawarnik

Ignored.

The Artist

@Sawarnik really? :'(

Sawarnik

6:02 PM

@UserX Yup :D

Balarka Sen

@UserX 1/1 is teh only rational with numer + denom = 2. 2/1 and 1/2 are the ones with denom + numer = 3.

The Artist

@Sawarnik but i seriously stopped

Ice Boy

@TheArtist you wanna troll me now?

3 mins ago, by Ice Boy

@TheArtist I said YOU ARE A TROLL

The Artist

@Ice Boy I stopped coz I'm getting ready to go :p

Daniel Fischer

@G.T.R Yes, in a uniform space, the closure of a set is the intersection of all of its uniform neighbourhoods.

Balarka Sen

6:03 PM

so the list will be 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, etc. @UserX

The Artist

@Ice Boy maybe later....gtg now :)

Ice Boy

see ya pal

Sawarnik

@IceBoy You should ignore him too.

Ice Boy

@TheArtist I'll be here if you wanna try it on me

The Artist

@Ice Boy tell that you won't ignore me ;)

@Ice Boy , sure sure .....bye for now

Ice Boy

6:05 PM

later

Balarka Sen

What the hell am I seeing about Sawarnik and Anastasiya?

Sawarnik

@BalarkaSen Just nonsense.

UserX

@BalarkaSen They're getting married.

Balarka Sen

@UserX Oh. I should congratulate.

UserX

Thanks for clearing that question out too.

Ice Boy

6:07 PM

just some troll's attempt

Balarka Sen

@UserX NP

Sawarnik

@UserX Don't you think it would be illegal?

@Nick Why did you flag once? :O

I have only flagged once too.

UserX

@Sawarnik why would it be illegal?

Sawarnik

@UserX We are 14.

Balarka Sen

@UserX underaged wizardry

Sawarnik

6:12 PM

@BalarkaSen I am going to leave the 14 club in less than a week now :P

Leave you with Anastasiya :P

UserX

Is $\log(e+\pi)$ irrational?

Sawarnik

And hey @Balarka How was the Diwali? :)

@UserX Nice question :D

Balarka Sen

@UserX Open problem, AFAIK

Sawarnik

I assume you didn't do anything except

locking yourself in a room.

Balarka Sen

@Sawarnik math, yes.

UserX

6:17 PM

If we prove it's irrationality, can we use the Gelfond–Schneider theorem to prove the trascedence of $e+\pi$?

Balarka Sen

no, it's open

Sawarnik

@BalarkaSen haha lol

UserX

I know it's open. But is it equivalent to showing that e+π is a trascedental number?

Balarka Sen

@UserX no, much more than that

it's stronger in fact

UserX

Why?

Sawarnik

6:19 PM

Last ping, @Anastasiya-Romanova sorry, whatever that artist troll did [neglecting the small probability that you are artist yourself]. just ignore whatever he said :))

Balarka Sen

$log(e)$ is algebraic, even though $e$ is transcendental

Sawarnik

@BalarkaSen :D

UserX

@Sawarnik stop being 14. What are you afraid of? That an online shy asian girl will know that you are into her? So what?

Sawarnik

@UserX and you stop too.

@BalarkaSen Why don't you celebrate a little? Its fun :)

Balarka Sen

@Sawarnik BS

Sawarnik

6:23 PM

BS?

Random Variable

@robjohn I was using the fact that $$\int_{-\pi}^{\pi} \frac{\sin nx}{\sin x} \ dx = \int_{-\pi}^{\pi} \text{Im} \frac{e^{inx}}{\sin x} \ dx.$$ But then I moved the $\text{Im}$ outside of the integral and said that $$ \int_{-\pi}^{\pi} \text{Im} \frac{e^{inx}}{\sin x} \ dx = \text{Im} \ \text{PV} \int_{-\pi}^{\pi} \frac{e^{inx}}{\sin x} \ dx.$$ But that seemingly doesn't make any sense, even though things somehow work out from there.

UserX

Let $\Bbb M$ denote the set of irrational numbers. Is $\Bbb M$ dense in $\Bbb R \setminus M$? I can't formulate this correctly.m.

Nevermind the last question. Fucked up my thoughts.

Sawarnik

:P

BS?

Balarka Sen

@UserX Huh?

Irrationals aren't dense in rational! They don't even overlap!

UserX

I know that's not what I had in mind

The Artist

6:32 PM

I am sorry if anyone got annoyed :) especially @Anastasiya-Romanova and @Sawarnick :) I was just joking :)

Won't repeat it again :)

Ice Boy

@TheArtist Do you want to talk about math or not?

The Artist

@Ice Boy , I do . But have to apologize :/ I didn't think he would tkae it serious :/ can u tell that i Told sorry ?

Daniel Fischer

@RandomVariable If you're extra nit-picky, you need to take the principal value before you move the $\operatorname{Im}$ out of the integral. But meh. Everybody does two steps of that sort at once. It makes total sense.

Ice Boy

@TheArtist the proof of that will be shown in your future behaviour

The Artist

@Ice Boy sure :p

@Ice Boy ok so...oh btw is it allowed to talk mecjanics/applied math here? Coz most people think it's phyiscs

Ice Boy

6:49 PM

The h Bar

General chat for Physics SE (physics.stackexchange.com). For M...

7

16

It is more on topic here^

Random Variable

@DanielFischer It doesn't make sense to me from the definition of the Cauchy principal value since $\frac{e^{inx}}{\sin x}$ not only has a simple pole at $x=0$, but also at $x=-\pi$ and $x=\pi$.

Semiclassical

@balarkasen: fyi, i managed to track down the connection i was rambling about yesterday re: Cantor sets, dynamical systems, and the 2-adic norm

Daniel Fischer

@RandomVariable You must take principal values not only at $0$ but also at $\pm \pi$, $$\lim_{\varepsilon\to 0} \int_{\varepsilon < \lvert x\rvert < \pi - \varepsilon} \dotsc$$

Semiclassical

if it sparks your curiousity (it might not) check out sections 1.5 and 1.6 of Devaney's text on dynamical systems

essentially, they use the 2-adic norm to describe how 'close' two different chaotic orbits of a quadratic map are

Daniel Fischer

@RandomVariable It might be clearer if you integrate say from $-\frac{\pi}{2}$ to $\frac{3\pi}{2}$, then you have two normal simple poles.

user112495

6:58 PM

I've got a matrix $A = \begin{pmatrix} 3 & -2 \\ 2 & -2\end{pmatrix}$. I've computed that $A^n = \dfrac{1}{3}\begin{pmatrix} 2^{n+2} + (-1)^{n+1} & 2(-1)^n - 2^{n+1} \\ 2^{n+1} + 2(-1)^{n+1} & 4(-1)^n - 2^n\end{pmatrix}$. But how do I find a polynomial $f$ such that $e^{tA}=f(A)$?

Semiclassical

sum it up by the dfn. of the matrix exponential?

though my approach to those kinds of problems is to write $A^2$ in terms of $A$ and $I$ by appealing to the Cayley-Hamilton theorem

user112495

@Semiclassical I think i'm partly confused as to what the 't' actually is.

@Semiclassical And f is a polynomial of degree less than 2.

Semiclassical

well, it depends on the context. in an ODE context i'd interpret as a $t$, and $e^{t A}$ as an operator that tells you the behavior at later times

in the same sort of way that, if you've got the scalar ODE $y'=-ky$, the factor $e^{-k t}$ converts $y(0)$ to $y(t)=e^{-k t}y(0)$

except more complicated b/c it's a matrix

if you don't have context, though, just treat it like a parameter

i.e. the thing you use to write $e^{t A}$ as a power series

but i really think you'll have an easier time if you think in terms of Cayley-Hamilton

user112495

@Semiclassical Ah, later on we use it to solve differential equations so that would make sense.

Semiclassical

right. it's really nice for that purpose, since as an operator it converts initial data (at $t=0$, say) into data for $y(t)$

The h Bar

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$Mathematics$ Mathematics