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Mew
10:00 AM
i thought taylor series was for differential equations
 
If you have an equation $y = P(x)$ for some polynomial $P$, then it can be inverted $x = P^{-1}(y)$ via Lagrange-Burmann expansion (google it). It's just a special kind of Taylor series.
 
Mew
can you show me how to do it in the case of my polynomial?
 
That'd be tedious :P
 
Mew
lol
aww
 
But it can be done.
 
10:02 AM
suppity sup
 
Mew
hello banana
 
heya
 
@BalarkaSen if what I think is write according to me all subspaces of R^n are the hyperplanes in R^n-1,hyperplanes R^n-2 , ........planes in R^3,lines going through the origin in R^2 an all lines and all of this which is going through the origin
 
:21212023 That would give you the previous integral, but I'm sure you know that.
 
moar category theory today
 
Mew
10:02 AM
did you know polynomials of any degree are solvable bananana
 
@robjohn Yeah :D
 
@Rememberme Well, you have to prove that there aren't any other subspaces.
 
@BalarkaSen have u heard of the category G-Sets whose objects are functors and whose morphisms are natural transformations?
 
How did you propose to show that?
 
$$
\begin{align}
\int_0^{\pi/2}\frac{\mathrm{Li}_2(-\sin(t))}{\sin(t)}\,\mathrm{d}t
&=\frac{\sqrt\pi}2\sum_{n=1}^\infty\frac{(-1)^n}{n^2}\frac{\Gamma\left(\frac{n}2\right)}{\Gamma\left(\frac{n+1}2\right)}\\
&=-\frac{\pi^2}8\log(2)-\frac7{16}\zeta(3)
\end{align}
$$
 
10:03 AM
@iwriteonbananas nah. what use would it be of?
 
i have no clue
 
@Chris'ssis I just need to show the last sum
 
@robjohn Yeah.
 
@Mew careful : "solvable" is a vague word.
 
@robjohn I found very diffficult the variant without minus sign in numerator.
 
10:04 AM
you have to specify the extension in which you hope to solve it.
 
Mew
nope, If I'm vague, noone can say i'm wrong
 
"hope to solve it" sounds very much like having to guess
 
Mew
the more specific one is, the more one is punished for small errors
 
By "analytic solution", I presume you mean a solution in $\Bbb C[[x_1, ..., x_n]]$
 
Mew
how do i solve this:
$x^7+7ax^5+14a^2x^3+7a^3x+b = 0$
i mean
x^7+7x^5+14^2x^3+7^3x+b = 0\,
$x^7+7x^5+14^2x^3+7^3x+b = 0$
 
10:08 AM
septics are very complicated. i can show you my method for quintics, say. or even sextics.
 
@BalarkaSen don't you have to study Galois theory in order to solve these equations
 
would i get rich and famous if i managed to find an algorithm for solving this polynomial without any guessing?
 
@Rememberme not really, no
@pZombie no : it has already been done
 
you don't do maths for becoming rich and famous @pZombie
 
says who?
 
Mew
10:10 AM
pZombie, an algorithm already exists for this polynomial
In mathematics, a septic equation is equation of the form where a ≠ 0. A septic function is a function of the form where a ≠ 0. In other words, it is a polynomial of degree seven. If a = 0, then it is a sextic function (b ≠ 0), quintic function (b = 0, c ≠ 0), etc. The equation may be obtained from the function by setting y(x) = 0. The coefficients a, b, c, d, e, f, g, h may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. Because they have an odd degree, septic functions appear similar to quintic or cubic function when graphed, except...
 
You do maths for enjoyment and the happiness that you get after solving some questions....its an art not for sale @pZombie
 
It's a De-Moivre septic, I guess.
 
don't get me wrong. I love knowing maths. It's the learning which i hate
 
Mew
a true lover of mathematics enjoys learning more than knowing
 
@Remember First do it for R^2. Show that there are no proper subspaces of R^2 other than the one-dimensional ones.
 
10:12 AM
@BalarkaSen few days ago I just read about something called the Ulan's spiral is it related to the Borusk Ulam theorem in topology?(its just out of curiosity I have read no topology )
 
no, it's not.
 
Mew
what is your favourite area of mathematics?
 
@Mew it varies for me. right now, it's algebraic topology.
 
Mew
For me it is probability and statistics
Are there any applications of algebraic topology?
 
is there something like a holy grail for mathematics similar to the theory of everything for physics?
 
10:14 AM
For R^2 it only has to be lines passing through the origin and the trivial one
 
@Rememberme proof?
 
Mew
yes pZombie
 
you're stating stuff without proving them
 
Mew - you know the question
 
certainly lines through origin are proper subspaces of R^2, but how do you prove that there aren't anything else?
 
Mew
10:16 AM
The holy grail of mathematics is to find the pattern of prime numbers
 
@Mew applications in what sense?
 
Mew
As in, economic utility
or utility outside of mathematics itself
e.g. engineering, medicine, technology, physics
 
Well, there are a lot of applications in physics, they say.
 
Mew what would happen if one was to find the pattern of prime numbers?
 
Mew
But I wonder how many physicists really understand algebraic topology
 
10:17 AM
$$
\begin{align}
\int_0^{\pi/2}\frac{\mathrm{Li}_2(\sin(t))}{\sin(t)}\,\mathrm{d}t
&=\frac{\sqrt\pi}2\sum_{n=1}^\infty\frac1{n^2}\frac{\Gamma\left(\frac{n}2\right)}{\Gamma\left(\frac{n+1}2\right)}\\
&=\frac{3\pi^2}8\log(2)-\frac7{16}\zeta(3)
\end{align}
$$
 
Mew
i'm not meaning to bag it out, I studied some basic topology and enjoyed it, but I was just curious if it had external use
but i understand the use of mathematics is often not discovered until centries afterwards
 
If I take a line which is not passing through the origin it won't have the [0,0] vector and if it dissent have something like that vector it won't be called as a subspace is what I think...
 
Mew
@pZombie, it would change the world as we know it
 
@pZombie he would be the Mozart of mathematics
 
Mew
very treu
 
10:19 AM
@Rememberme you're just proving that lines not passing through the origin are not subspaces. but what if i claim that there is some "curve" passing through the origin? how would you prove that there would not be such a subspace?
 
Mew
@pZombie, you can be very rich if you discover the pattern of sha random number generators
 
Ok for curve.....I can take take and stretch it form a line :p
 
that's nonsense
 
I am surprised mathematicians haven't found a general algorithm to solve any kind of mathematical problem, including finding the pattern of primes. This is what i would consider the holy grail
 
10:21 AM
I knew this was coming
 
@Remember i am pretty sure what i mention is in Hoffman-Kunze. you haven't studied vector spaces carefully.
 
Mew
Yes the pattern of primes has baffled mathematicians for millenia
 
stop claiming you understand something if you don't.
 
I was just joking let me think about it
I have told you for a line..... I have to think about for a curve
 
even that wouldn't suffice.
 
Mew
10:23 AM
how old are you pZombie
 
i don't give out any personal data on the internet. TPTB could use it to track me down
 
Why you mean for a line....
 
@Remember you have to do it in general for a set.
there are a lot of subsets of R^2. how would you know that one of them ain't a subspace?
 
Mew
no start with a circle
 
Ok so let me just generalize it ...
 
10:24 AM
Mew - how old are you?
 
Mew
15
 
doubt it
 
Mew
wat do u mean?
y don't u belieave me
 
you seem to know too much
 
Mew
oh ty lol
yea i'm a little nerdy
 
10:27 AM
And also for a subspace to be true it has to be close under multiplication and addition
 
yes
 
@robjohn How you got that result?
 
@BalarkaSen if $G$ is a group, what is meant by the $K$-linear representation of $G$?
 
Barlaka is probably 27
 
@pZombie balarka is 14
 
10:28 AM
Bananas is around 20
 
I am 14
 
Mew
Yes Barlaka is 14
 
stop talking about that nonsense
 
Does that suffice @BalarkaSen
 
Mew
man i'm the oldest one here
 
10:29 AM
do math
 
Mew
i feel so old
 
Me @iwriteonbananas
 
Mew
at age 15
 
@iwriteonbananas i guess a morphism $G \to GL_n(K)$. Dunno.
 
Mew
I was 14 when I finished my degree
Are you doing a phD balarka?
 
10:30 AM
loads of crap-talking going around, i guess
 
@BalarkaSen also for the other subsets
 
yeah its unbearable today....too much background noise
 
i didn't see your proof, @Remember
 
Mew
how so
 
10:30 AM
im gonna get off the internet
 
Mew
chat room isn't for questions, that is what the main site is for
 
me too
 
Wait
 
see ya, @iwriteonbananas
 
Mew
Do your own homework
 
10:31 AM
Wait @BalarkaSen
 
you killed the chat Mew
 
@Remember write up a sensible proof, then ping me
 
Mew
lol
oops
 
Ok fine
 
Mew
IS balarka your teacher
 
10:32 AM
i wish, i seriously need a good math teacher
 
Mew
Man I can teach yoi
I know all maths
when you know everthing it's boring
 
now that's a bold statement
 
Mew
but when I teach, I can relive the excitement of someone learning for the first tiem
 
i need the maths to understand GR and QM
 
Mew
Ah
 
10:34 AM
@robjohn OK, I got the point.
 
Mew
well the maths required for GR is diffferent to QM
For GR you need a thorough understanding of vectors, and then tensors
for QM, you need to know linear algebra and differential equations
 
Mew you understand GR?
 
Mew
you also need to know the basics of functions and goemetry
yeah
 
quite an achievement at 15
 
Mew
general relativity is pretty simple
ty
 
10:36 AM
you also understand QM?
 
Mew
yes quantum mech is very simple
although it can be hard to wrap your head around at first
 
is a photon a wave packet?
 
Mew
since it is counter intuitive and very different to classical mech mathematically
yeah kind of
the behaviour of photons is modelled by quantum electrodynamics
that is beyond quantum mechanics
 
can an empty universe contain anything other than a single photon in it?
 
Mew
QED is harder than QM
I'm still learning QED
also known as quantum field theory
Quantum mechanics involves studying electrons and other particles
Quantum field theory studies the quantum mechanics of photons and forces
The holy grail of physics is uniting Quantum Field theory with General relativity
QFT is harder than GR to learn and is the pinaccle of physics education
pZombie, your question relates more to philosophy. If it had stuff in it, it's not really "empty" is it
 
10:40 AM
i said "other than", i did not say it is empty
 
Mew
you said can an "empty universe" contain ...
empty universe contains nothing
to learn physics you must master logic first
this is your first lesson
 
alright then, a universe containing nothing other than a photon
and of course be full of vacuum
 
Mew
yes it is not theoretically impossible
 
isn't the wave packet a photon is essentially a creation of other waves?
 
Mew
wtf
i have no idea what you just said
 
10:43 AM
and does it's existence depend on the continuation of those other waves?
 
Mew
no
a photon is a wave
 
it's a wave packet i believed
 
Mew
no other waves are requried other than the photon itself
a wave packet and a wave are the same thing
the "packet" just means the wave's distribution is bell curve shaped
in that there is a high probability of finding the photon at the peak of the bell curve, and a lower probability at the tail of the bell curve
 
then what is this about youtube.com/…
 
Mew
see the purple graph in the back right?
 
10:46 AM
yes, i am not blind
 
Mew
that purple graph shows the "wave packet"
 
not
 
Mew
in reality the wave packet continues along the +x axis infinitely
 
he explains how the wave packet emerges
 
Mew
and -x axis infinitely
 
10:46 AM
no, that's a planar wave, he also explains
 
Mew
no
the planer wave is the red wave
the wave packet is the purple wave
 
don't you see the arrow even pointing to "plane wave" ?
 
Mew
Look at the picture here
that is a plane wave in 3d
soryr i mean packet
above is a 2d picture of a wave packet
A wave packet is simply a wave function that appears to be localised in space
the wave does cover all space, but the hight of the wave is significantly small away from the localised position
 
"if we take the superposition of many plane waves.... we can in fact obtain a single wave which is localized... a result of the superposition of many plane waves .... known as a wave packet"
part of what he explains if you listen to the video
 
@robjohn It was ridiculous saying that is very hard. I just did a horrible mistake in my calculations. Shame on me.
 
Mew
10:50 AM
yes that is correct pzombie
however note a single photon isn't = a wave packet
But rather a single photon usually exists as a wave packet
This is kind of technical, but you don't need many photons to get a wave packet
Even one photon itself can be in a superposition of many states, giving rise to a guassian shape
A photon can interefer with itself
If you had just one photon in the whole universe, the photon woulnd't be localised, but it would be spread over the entire universe uniformly as a plane wave
 
we wouldn't call that a photon then really, would we?
 
Mew
well in QFT we still would call it a photon
 
@robjohn I'm done with all stuff (after making all the proper corrections).
 
Mew
that is why i said photons usually are packets
 
@robjohn hence I immediately also get the result for $$ \int_0^{\pi/2}\frac{\mathrm{Li}_2(\sin^2(t))}{\sin(t)}\,\mathrm{d}t$$
 
Mew
10:57 AM
Lithium is 3 not 2
\int_0^{\pi/2}\frac{\mathrm{He}_2(\sin^2(t))}{\sin(t)}\,\mathrm{d}t
$\int_0^{\pi/2}\frac{\mathrm{He}_2(\sin^2(t))}{\sin(t)}\,\mathrm{d}t$
That's better
 
@robjohn moreover, I also derived the cute version $$\int_0^{\pi/2}\frac{\mathrm{Li}_2(\sin^2(t))}{\sin^2(t)} \,\mathrm{d}t=2 \pi( 1-\log(2))$$ that is far easier than the previous versions.
Although it might look at first sight less friendly.
 
11:13 AM
@balarka please come
 
quick galois theory question, should be easy but i'm being stupid
Let $p$ be prime of the form $n^2+1$
The Galois group of the simple extension $\mathbb{Q}(\sqrt{p+\sqrt{p}})$ turns out to be isomorphic to $C_2 \times C_2 $
generated by $f: \alpha \mapsto - \alpha$ and $g: \alpha \mapsto n \alpha^{-1} $
I want to see the correspondence now,
It's easy to see $\mathbb{Q}(\sqrt{p})$ is the subfield fixed by $<f>$
For some reason I can't see what the subfield fixed by $<g>$ is
I know it's also of degree $2$ etc by the fundamental theorem
 
@JC Why is D_{2n} nilpotent iff n=2^i for some i>1?
I have tried so much shit
Shapes and crap
 
11:33 AM
3
Q: Is the dihedral group $D_n$ nilpotent? solvable?

takecareIs the dihedral group $D_n$ nilpotent? solvable? I'm trying to solve this problem but I've been trying to apply a couple of theorems but have been unsuccessful so far. Can anyone help me?

@Incurrence
 
@SanathDevalapurkar is pretty smart
 
OK i messed up with my galois stuff
so ignore that Q
should be a $\sqrt{p}$ tagging along in places
 
11:55 AM
@BalarkaSen I can't get this damn $d_{2n}$ to be nilpotent iff if $n=2^{i}, i\gt 1$
I am using the presentation $\langle r,s| s^2=e,r^n = e, srs=r^{-1}\rangle$
 
@Incurrence the question above provides a proof using induction
 
@BalarkaSen Honestly, I didn't understand it
and I am surprised no simple proof exists
but I'll be right back with foods
 
what did you not understand?
 
Let p be an odd prime dividing n
so we want it to be a prime less than our 'modulo'
and that n is a multiple of p
OK done eating
Ok
 
well, we are assuming that some odd prime p divide n, and we are contradicting that which implies nilpotency.
what's the problem with that?
 
12:07 PM
@Incurrence you must have ate a single peanut
 
@ᴇʏᴇs Power bowl of cocoa pops for first time in years
@BalarkaSen Ahhh I see
 
Did you even chew @Incurrence
 
@BalarkaSen Well I sort of see that
It means that either $p$ is an even prime, $2$, or it isn't divisible by a prime
Which is impossible
So we have divisible by $2$
 
yes.
 
Which means that
 
12:14 PM
Ok I think the Galois Group is actually $C_4$ now haha
I'll check
 
galois group of what?
 
$\mathbb{Q}(\sqrt{p+\sqrt{p}})/\mathbb{Q}$ where $p$ is a prime of the form $n^2+1$
 
that looks right. you just need to check for elements of order $4$ in your galois group.
if there is one, it's $\Bbb Z/4$, it there ain't one, it's $\Bbb Z/2 \times \Bbb Z/2$
 
yeah found one
 
thumbs up
 
12:19 PM
Because of a small calculation error originally I had it as the klein four, but then the correspondence messed up completely so I knew something was up
alright
going off
cya!
 
What? The exponential map for real matrices $M_{n\times n} (\Bbb R) \to GL_n(\Bbb R)$ is not surjective?
How would I find some example that isn't mapped to?
 
i thought it was surjective.
 
Apparently not
 
The determinant of real valued matrices are positive
 
So we take two eigenvectors with negative eigenvalues on a 2x2?
 
12:26 PM
ahh
i get it
@Incurrence note that if $\lambda$ is an eigenvalue of $A$, then $e^\lambda$ is the eigenvalue of $e^A$.
 
@DiscipleofBarney wut?
@DiscipleofBarney $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$
 
Under the map, the matrices have positive det
 
12:47 PM
What do you think, does power series reversion have anything to say about the zeros of a polynomial?
 
1:12 PM
I don't understand power series expansion.
 
1:45 PM
Hello!! Is someone of you familiar with Cryptography??
 
@Gigili ok, can you repeat that again? Am online now.
 
2:09 PM
@TedShifrin I heard the MIT was very selective. Is there not an entrance exam there ? I think this kind of exams is the fairest tool to recruit students based on their skills, since everybody gets to be on an equal footing
 
@LeGrandDODOM, no MIT does not have an entrance exam.
 
2:26 PM
Although there is no entrance exam, there is an exit.
 
2:41 PM
@MaryStar It might be best to state the question. If someone can answer it they will.
 
@robjohn Hi, I am still alive.
 
3:01 PM
@WillHunting That is good news
 
3:20 PM
$e^{d/dx}f(x)=f(x+1)$
3
 
@Semiclassical my research led me to some amazingly strinking results, like the one above (I don't use elliptic functions to evaluate the one below). $$\int_0^{\pi/2} \frac{1}{1+\sqrt{\sin(x)}} \ dx$$
I wonder if anyone managed to do this before (with basic tools) ...
 
@Incurrence are you free I have an idea for a proof I need to show you
 
@Semiclassical I meant below (not above)
 
@Chris'ssis Hello Superhuman! Hope your book is going well...
 
@WillHunting Hi. Well, yeah, it works fine. Now I'm a bit more involved in some research (but indirectly this can also come to my book). :-)
 
3:27 PM
@Chris'ssis That day you said you have 10,000 problems. I said you can write 20 books. I was serious. =) 1 book can have 500 problems. And you will be rich. =)
 
@WillHunting I have much more than that, but not all problems are worth to be published. The readers are sensitive, you need to know how to give them your best. I still learn doing that.
 
@columbus8myhw What is your formula supposed to mean?
 
@WillHunting What is your question supposed to mean?
 
@ᴇʏᴇs I don't know. He just typed it out of nowhere.
@Chris'ssis You sound like a very experienced author. =)
 
@WillHunting Not really. Excepting writing some training courses (50 pages), I never attended a book before.
 
3:35 PM
@Chris'ssis I hope to wrote 20 books one day covering all major branches of math. =)
 
@WillHunting Then you should start today. ;)
 
hi @iwriteonbananas
 
3:52 PM
@ᴇʏᴇs I am going to take a nap, see you in my dreams.
 
good morning guys
 
Hi @BalarkaSen I have got a brilliant idea
 
fire it at me.
morning @Mike
 
@BalarkaSen hi
 
morning
 
3:58 PM
im going crazy
hey mike
 
why is there no natural transformation Sym $\to$ Ord ?
 
blergh. category theory.
 

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