Mathematics - 2018-02-05 (page 1 of 6)

Antonios-Alexandros Robotis

00:27

hi @Daminark

Daminark

00:44

Yo

@Antonios

Dom

Hi folks, I wonder if anyone could answer a pretty simple math question for me please

Antonios-Alexandros Robotis

perhaps :P

Daminark

I'm pretty sure someone could

Dom

Given that where 'ax^2 + bx + c = 0', if the discriminant (D = b^2 - 4ac) is less than 0 there are no solutions, equal to 0 and solution is x = -b/2a, and if greater than 0 the solution is either of x = (-b-sqrt(D)) / (2a) and x = (-b+sqrt(D)) / (2a) ---------------- does this hold true for any value (instead of 0), or only where ax^2 + bx + c equals 0?

Antonios-Alexandros Robotis

Well, your problem can be reduced to the following:

$ax^2+bx+c=d\iff ax^2+bx+(c-d)=0$. So, just use the previous criterion on the new equation.

So, basically yes, depending on what you mean.

Dom

00:51

hmm, thanks, I wonder if that'll work

I need to think about it for a few moments

Semiclassical

One way to get a handle on what's going on is to think about the graph of a quadratic

Antonios-Alexandros Robotis

No problem. Feel free to ask more if you need.

Dom

I'm applying it to a physics equation for projectile trajectory, and trying to refactor the equation to figure out how long it will take to collide with a boundary on the x axis

I figured I could translate the path over to move x to 0, which is what it looks like that would do

Semiclassical

it's a parabola which either opens up or down. If it opens up and the x-axis is above/below the vertex of the parabola, then there's 2 / 0 real solutions

Dom

but it's complicated and my maths are out of practice

Semiclassical

00:53

if you change the value you're setting the quadratic equal to, that's basically the same as shifting the x-axis up/down

so if said axis moves across the vertex, you'll change the number of solutions

Dom

@Semiclassical yeah all that matters is the trajectory if I can reverse whatever changes I make to make the calculation

Antonios-Alexandros Robotis

hey @Daminark what's the tikzcd code for putting two arrows between the same two nodes so they don't overlap

Daminark

Not sure but do you know if the tikz editor online?

Antonios-Alexandros Robotis

nope, but i figured it out

Y_1\arrow[rr,"f",yshift=.85ex]&&Y_2\arrow[ll, "g"]

this will do it

Daminark

God

Antonios-Alexandros Robotis

00:59

the only exotic bit is yshift

Dom

@Antonios-AlexandrosRobotis Oh yeah it's coming back to me now, thanks

Antonios-Alexandros Robotis

No problem :)

Dom

that was a pretty basic question :D

Antonios-Alexandros Robotis

better to know than not to know :)

Narcissus jewel

01:15

@Antonios-AlexandrosRobotis Shouldn't you just bend the arrows?

Antonios-Alexandros Robotis

I think it looks better with straight arrows in this case

but that's also an option :P

Kasmir Khaan

Hi chat

Antonios-Alexandros Robotis

hi @KasmirKhaan

Narcissus jewel

@Antonios-AlexandrosRobotis Actually that looks really nice

Kasmir Khaan

I like Antonios , he is a nice guy, never let me down when it comes to say hi :D

Antonios-Alexandros Robotis

01:16

:P

Narcissus jewel

Hi @KasmirKhaan

Kasmir Khaan

So transparant Jewel -.-

but hi :)

Antonios-Alexandros Robotis

@Narcissusjewel yeah it can look nice. sometimes bending is better

Kasmir Khaan

anyways, any of you folks familiar with rep theory of finite Groups?

Narcissus jewel

@KasmirKhaan Not sure what you mean

Antonios-Alexandros Robotis

01:17

a matter of taste, I guess.

Kasmir Khaan

Serre book :D

Narcissus jewel

@KasmirKhaan What about it?

Antonios-Alexandros Robotis

I am doing a little bit right now @KasmirKhaan but I'm only learning.

Kasmir Khaan

I need help with some questions from that book

It is so hard -.- and its bad that we did not do linear algebra properly in my uni

Narcissus jewel

p151?

Kasmir Khaan

01:18

@Narcissusjewel no second , third and forth exercices :D

Dom

Is there a formula to find the length of the quadratic curve from origin to a specified vector/set of coordinates on the graph?

Kasmir Khaan

@Narcissusjewel page 12 !

Dom

or from an arbitrary point, I chose origin to keep it simple

Narcissus jewel

@KasmirKhaan Well tell me what you've tried for each and I'll help you

Or do so in an hour, since I shouldn't get distracted from my work yet

Antonios-Alexandros Robotis

@Dom tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx

Kasmir Khaan

01:22

@Narcissusjewel well thanks for the offer, keep working on your thing, my main problem is not understandign the defintions well enuf ><

we meet again in 1 hour @Narcissusjewel

Antonios-Alexandros Robotis

which defns @KasmirKhaan

Kasmir Khaan

hmm many tbh

character, tensor ,

Antonios-Alexandros Robotis

okay, well what do you understand about characters thus far

Kasmir Khaan

Sym^2 and ALT^2

well it is a trace

Dom

@Antonios-AlexandrosRobotis thanks, it's a lot to consume

Antonios-Alexandros Robotis

01:24

no problem @Dom, but that is the best way

okay so, the basic setup is we have a hom. $\rho: G\to \text{GL}_n(\mathbf{C})$ -- at least in the basic theory.

Right?

Dom

unfortunately I get stuck on a lot of symbols and I don't use them often enough to memorise or remember them

Kasmir Khaan

Yes

Dom

It's hard to even describe a lot of them and I don't think they copy-paste in most cases

Kasmir Khaan

that is the defintion of a representation

for each g in G, we associate with it an element of GL(V)

Antonios-Alexandros Robotis

Yep. All we're doing with a character is we're sending $g\mapsto \text{tr}(g|_{V})$.

Kasmir Khaan

01:26

bad notation but GL(n,K)

Antonios-Alexandros Robotis

That is, $\chi(g)$ is exactly that trace.

Kasmir Khaan

but this trace does not depend on basis?

Antonios-Alexandros Robotis

Trace is invariant under change of basis.

Kasmir Khaan

I have some gaps in linear algebra

aha so it is well defined nice

Semiclassical

a change of basis amounts to $M\mapsto A M A^{-1}$

Kasmir Khaan

01:28

okay say taht we did that for all elements of G

Antonios-Alexandros Robotis

Trace is invariant under cyclic permutations of the argument: $\text{tr}(ABC)=\text{tr}(CAB)=\text{tr}(BCA)$.

Kasmir Khaan

since we only work with finite groups

Antonios-Alexandros Robotis

In particular, it is invariant under change of basis.

Kasmir Khaan

thanks that is very usefull :D

Antonios-Alexandros Robotis

np.

Kasmir Khaan

01:29

but once one have the character of any element

Semiclassical

it shouldn't really be a surprise: the eigenvalues of a matrix shouldn't depend on the basis, and the trace gives the sum of the eigenvalues

Kasmir Khaan

g in G

by that i meant, g---> a in GL(V)

we get few values

2,-2,0,4 say

what does that say to us?

@Semiclassical Semi the last thing i want to hear is that something is obvious -.- i hate that Word now

Antonios-Alexandros Robotis

well, I think the "big deal" with characters is that representations (in this specific context) are uniquely determined by their characters.

Semiclassical

hey, just because it's not surprising in retrospect doesn't mean it's obvious

Kasmir Khaan

hmm

Antonios-Alexandros Robotis

01:34

I would recommend checking out the section in fulton & harris. It's fairly clear.

0celo7

@EricSilva [F4, Section 3.2.29]...that can only be Federer

Kasmir Khaan

@Semiclassical @Antonios-AlexandrosRobotis thanks guys, i Think I should keep trying on my own and come up with good questions if needed for later :D

Antonios-Alexandros Robotis

okay :) good luck

Kasmir Khaan

@Antonios-AlexandrosRobotis what section ?

0celo7

phd level linear algebra above

Antonios-Alexandros Robotis

01:35

section/lecture 2

Kasmir Khaan

hmm is this online ?

Antonios-Alexandros Robotis

probably lol

Kasmir Khaan

i meant is it a video or a book

lecture 2?

Antonios-Alexandros Robotis

the book is divided into "lectures"

but it is in fact a book

Kasmir Khaan

haha makes sense =P sorry for being slow :D

Antonios-Alexandros Robotis

01:36

I introduced the ambiguity :P

Adeek

01:56

yo @Antonios-AlexandrosRobotis do you know little bit about Ext functor ?

for groups

for simplicity

Antonios-Alexandros Robotis

not much, I'm afraid.

Adeek

Do you know why Ext^i(A,G) = 0 for i >= 2 ?

oh wait I get it

@Antonios-AlexandrosRobotis do you want to know why ?

Antonios-Alexandros Robotis

oh hi what's up?

sure

Ted Shifrin

howdy @Antonios, Karim

No homological algebra for Ted, thanks.

Adeek

@TedShifrin Hi :D Yeah homological algebra gives you magical things

Xander Henderson

02:07

how about some fractal homology?

Ted Shifrin

Nope, none of that, either.

Adeek

@Antonios-AlexandrosRobotis so any free group is project

projective

Xander Henderson

Look, it isn't that hard---it's just a graded group!

Of course, the grading is $\mathbb{R}$, but that shouldn't be too bad to work with, right?

Adeek

so once we have Ext^2 it becomes free so it is projective and so the homology vanishes

Xander Henderson

I mean, it could be worse...

Adeek

02:08

@Antonios-AlexandrosRobotis

@TedShifrin hahaah do you want to hear funny story ?

Ted Shifrin

Nope.

Antonios-Alexandros Robotis

hi @TedShifrin

Adeek

@TedShifrin So today I was sitting eating while thinking about some sequence (diagrams). I was saying to myself oh yeah you apply some yoga to the sequence and by chopping off at this end. My wife tells me what yoga and was telling me how mathematicians have weird phrasing

haha

@TedShifrin yeah my jokes don't make sense to me as well :D

lolz

Antonios-Alexandros Robotis

@TedShifrin Meeting with the professor of the algebra course tomorrow [she couldn't make friday]. Trying to think of how to word what I need to say. I'm thinking that maybe I should hold a few short "review" lectures.

Adeek

@Antonios-AlexandrosRobotis I just discuss ideas with my professors

I don't discuss like rigorous things

Antonios-Alexandros Robotis

02:22

this is for teaching

Adeek

oh

0celo7

@Adeek wot

Adeek

@0celo7 WOOT WOOT turn down to wh!t

haha

that music video always make me laugh

Akiva Weinberger

goodreads.com/quotes/…

> "I don't know how many of you have ever met Dijkstra, but you probably know that arrogance in computer science is measured in nano-Dijkstras."

Also:

@Semiclassical smbc-comics.com/comic/2014-12-06

0celo7

@AkivaWeinberger the witcher 3?

Akiva Weinberger

02:36

??

0celo7

you don't know what the witcher 3 is?

Akiva Weinberger

Oh apparently it's a game

with a character named Dijkstra

Xander Henderson

You know, when someone says "Dijkstra", some obscure video game is not the first thing that comes to mind...

Akiva Weinberger

Depends on the person, I'm guessing

In any case, Edsger Wybe Dijkstra (Dutch: [ˈɛtsxər ˈʋibə ˈdɛikstra]) was a computer scientist

The "ij" sounds like the "ij" in @Krijn

Joe Shmo

Hey y'all. Trying to work problem 2.8 here --http://www.math.cornell.edu/~sjamaar/manifolds/manifold.pdf

It's a bit cumbersome to write out.

In any event, I'm getting dg = f_1(x1,...xn)dx1 + f_2(0, x2, ..., xn)dx2 + ... + f_n(0, 0, ..., 0, xn) dxn. Why is that the same as alpha (as described in the problem)? I'm not taking advantage of the fact that alpha is closed, i.e. d alpha = 0 wherein lies the problem in all likelihood.

orbit-stabilizer

02:45

His name is fun to say @Akiva

Akiva Weinberger

Who, Dijkstra or Krijn

orbit-stabilizer

Edsger Dijkstra

0celo7

@XanderHenderson obscure?

my god

are you serious?

Semiclassical

@XanderHenderson ehh. I’m not sure one can call Witcher 3 ‘obscure’ among video games

0celo7

you're not sure!?!?

how about "it's literally insane to call it obscure"

Xander Henderson

02:54

?

Semiclassical

I was trying to be diplomatic :p

Akiva Weinberger

@0celo7 Wrong crowd

Antonios-Alexandros Robotis

I used to be quite into video games. Long before witcher 3 came out though, I imagine.

0celo7

@AkivaWeinberger what do mathematicians do if not play video games

@XanderHenderson it sold 10 million copies

Semiclassical

As of last March it had sold 10 million copies

Antonios-Alexandros Robotis

02:56

@0celo7 presumably other things :P

0celo7

@Antonios-AlexandrosRobotis How can a young male in 2018 not play video games

Antonios-Alexandros Robotis

Well not all mathematicians are young males in 2018. But, on the flip side, you can quite simply have other hobbies lol

orbit-stabilizer

I don't play video games

0celo7

wtf

why not?

Antonios-Alexandros Robotis

Actually, most of the friends I had in undergrad who were also math people didn't play video games.

Xander Henderson

02:57

@0celo7 is that a lot?

Semiclassical

TBH I don’t play video games either

0celo7

is this the twilight zone?

Semiclassical

I’m too cheap to pay for it

Akiva Weinberger

I recently bought Quantum Conundrum and then realized it's PC only

(I have a Mac)

Antonios-Alexandros Robotis

I don't play video games anymore because it doesn't feel like time well spent for me right now.

Akiva Weinberger

02:58

On the other hand it was like two dollars

Joe Shmo

no differential geometers here?

orbit-stabilizer

Because, I do other things? I like listening to podcasts and working out and sleeping. And I don't have time to do all that and school and play video games.

0celo7

@Antonios-AlexandrosRobotis it's definitely not time well spent, but that isn't the point

Semiclassical

That said, I do follow a decent amount of video games news

0celo7

@JoeShmo what do you need

Antonios-Alexandros Robotis

02:58

Not to cast judgement on other people – I just can't do it anymore.

Akiva Weinberger

I used to be really into Minecraft

I dunno, high school is a big time drain for me

Xander Henderson

Is Witchar 3 anything like Castlevania? 'cause that was a great game...

Joe Shmo

@0celo7
Trying to work problem 2.8 here --math.cornell.edu/~sjamaar/…

It's a bit cumbersome to write out.

In any event, I'm getting dg = f_1(x1,...xn)dx1 + f_2(0, x2, ..., xn)dx2 + ... + f_n(0, 0, ..., 0, xn) dxn. Why is that the same as alpha (as described in the problem)? I'm not taking advantage of the fact that alpha is closed, i.e. d alpha = 0 wherein lies the problem in all likelihood.

Xander Henderson

or is it more like Megaman?

Semiclassical

In genre, yes

0celo7

02:59

@XanderHenderson it's like Skyrim

and if you've never heard of that, we're done here

Xander Henderson

well, then I guess we are done

Antonios-Alexandros Robotis

LOL

Xander Henderson

Skrim sounds like a silly name for a game

0celo7

holy shit you're serious?

Akiva Weinberger

I do like video game music, but for that you just need YouTube and a good sound system

Xander Henderson

02:59

Megaman makes sense... you plan a MAN who is quite MEGA

0celo7

@XanderHenderson how old are you?

Akiva Weinberger

(Hm. If it's written for the NES, do you need a good sound system?)

orbit-stabilizer

Skyrim was meh

Transcript for

$Mathematics$ Mathematics