Mathematics - 2016-11-27 (page 2 of 11)

Jessy Cat

01:00

@Semiclassical, we could always extend it so that it is.

TheGreatDuck

and your cells consumed nutrients and oxygen

therefore

Hiro

Yeah but the amount of energy you lost isn't significant

Jessy Cat

But nothing else is told to us about $g(x)$ except that it is defined on $(0,L)$.

TheGreatDuck

your hand did work

no amount of energy "isn't significant"

tis still work

Semiclassical

Then how can you infer that it's even? That's a statement beyond (0,L).

Hiro

01:01

okay how about this

TheGreatDuck

just near little amounts of work

Hiro

You didn't do work in macroscopic level

not the microscopic level

Semiclassical

I mean, as a very simple example: Take $g(x)=0$ for $0<x<L$ and $g(x)=1$ everywhere else.

Danu

o/

Jessy Cat

@Semiclassical I posted a question about it on the main MSE page and no answers, but a couple of guys said pretty much the same thing you did. What can we say, then?

TheGreatDuck

01:01

i'll give you that one. ;)

Danu

Does anyone have any recommendations concerning representation theory of Lie algebras for undergraduates?

Jessy Cat

Well, couldn't you extend it so that it is even?

Semiclassical

I suppose so? I could be missing a counterexample, though.

Jessy Cat

Er, what can we say about $g(x)$ that is, @Semiclassical given that $g(x)$ is defined on $(0,L)$ and that integral is zero?

Semiclassical

But, really

Hiro

01:02

@TheGreatDuck If 1=0, prove that you're the pope ;)

Jessy Cat

@Hiro that deserves a star.

Semiclassical

If you're given $g(x)$ on (0,L), then you can always extend that to be even.

Just define $\tilde{g}(x)$ to be $g(|x|)$.

Jessy Cat

@Semiclassical this is what I said: math.stackexchange.com/questions/2032202/…

Could you take a look? I felt all confident for the like the first time ever on this assignment, and then I post just to make sure, and now I'm all muh.

Semiclassical

In order to say anything definitive about stuff outside (0,L) you need additional assumptions.

Now, you can probably say that certain things are permissible.

Jessy Cat

I think that's what those guys on there were getting at. But, it looks here like I'm really only interested in (0,L)

Semiclassical

01:05

But if someone just tells you "$g(x)$ behaves on (0,L) in this way"

then you can't say anything about it away from that interval.

Null

@Hiro let the pope be denoted by 1 and everyone else as 0. I am everyone else, therefore i am the pope.

2

Jessy Cat

Do I need to though in order to answer the question?

Semiclassical

Maybe not.

I think you could say that it's symmetric under $x\mapsto L-x$?

Though maybe that should be antisymmetric for one and symmetric for the other.

Jessy Cat

@Semiclassical would you be interested in posting an answer?

Semiclassical

Not really.

Jessy Cat

01:07

:(

Hiro

@Null Nice try, but that means that you along random people are the pope, so try to prove that you and only you are the pope.

Jessy Cat

I have a feeling nobody is going to.

Null

@Hiro books.google.de/…

basicly add 1 until we got 5=4^^

Jessy Cat

What about what the one guy said about how we need to know that $g(x)$ is even integrable on $[0,L]$?

Semiclassical

Tbh, I don't know. This gets into the formal details of Fourier series which I don't remember.

Jessy Cat

01:09

Beautiful.

Hiro

@Null the way you'd go about it would be that if 1=0, then 2=1 by adding 1 to both sides, and if the pope and I are two people, then the pope and I are one person, therefore I am the pope

Semiclassical

The 'soft' conclusion I"d draw is that the Fourier series in the first case contains only cosines which are $[0,L]$ periodic, and the second only sines.

Jessy Cat

Because I know that you need a whole bunch of criteria - Dirichlet's conditions - in order for the series to actually converge to $g(x)$, but nobody said anything about needing that. There are lots of Fourier series that don't converge to their respective functions.

Semiclassical

Right.

Null

@Hiro our analysis tutorer made that joke, forgot it :(

Jessy Cat

01:11

@Semiclassical, don't know if you're a Game of Thrones fan, but that's the moment when the Hound would say "Lots of expletives".

Hiro

@Null lol if you're in a logic class, the teacher will definitely bring it up

Jessy Cat

What is all of you guys' obsession with the Holy Father tonight?

@Semiclassical in any case, that's pretty much the same conclusion I came to. Unless somebody answers the question I posted, I'll stick to it, and maybe add on a bit about not being able to tell anything about the behaviour of $g(x)$ outside of the interval $(0,L)$, yadda yadda.

Danu

Hi @TedShifrin

Jessy Cat

And since TedShifrin won't answer any more of my questions since I'm cray cray :(

Danu

Do you have any opinions on good books for representation theory of Lie algebras?

Jessy Cat

01:15

He is like the god of PDEs.

Krijn

@Danu Fulton and Harris is the classic right?

I have a course on exactly that at the moment

Danu

Could be!

I'm looking for something to recommend to undergrad students

(is this even feasible?!)

Krijn

@Danu Fulton and Harris is probably very hard for undergrad students

At least, in my experience with the book

Danu

@Krijn Yeah, way too hard probably. I'm definitely not looking for that, it seems :P

Ted Shifrin

@Danu: I am a baby when it comes to representation theory. I love to recommend Curtis's introductory Springer book on Lie groups, but that doesn't fit your bill.

Danu

01:20

@TedShifrin I was looking at Stillwell's "Naive Lie Theory".

Ted Shifrin

Representation theory of Lie algebras is mostly (reasonably advanced) linear algebra. I would start by looking at Jim Humphreys's book.

Krijn

Is this a baby like when Balarka says that he's a baby in Algebraic Topology, or a proper baby?

Danu

Not really about representations though.

Ted Shifrin

But, still, it's probably graduate level.

Danu

@TedShifrin This is way too hard for my kids :\

Ted Shifrin

01:21

ROFL @Krijn

Danu

I remember struggling with it myself 2 years ago

Ted Shifrin

Why should kids be doing representation theory?

Danu

@Krijn No :P

@TedShifrin I just want some really basics

Because it's very useful in particle physics, to make sense out of the zoo of particles in the standard model.

Ted Shifrin

I do not know an appropriate resource.

Danu

And to inspire them a bit when they encounter the $\mathfrak{su}(2)$ algebra generated by angular momentum operators

Ted Shifrin

01:22

But this still requires a comfort level with linear transformations that many undergraduates do not have.

Danu

What does?

Ted Shifrin

What you just said.

Danu

Meh, physicists do a good job giving some intuition without any formal knowledge :D

The concept of operators acting on a space of states (Hilbert space) is very natural for physics students, so the basic idea of a representation comes rather naturally.

Ted Shifrin

Take a look at this.

Krijn

You paraphrased some really negative comments in a positive daylight there.

Jessy Cat

01:25

@TedShiftin Re: "Why should kids be doing representation theory?" Well, it's better than doing drugs, isn't it? ;P

Ted Shifrin

Unless it drives them to same, @Jessy :)

Danu

@TedShifrin Oh, neat!

@TedShifrin hahahaha

DESPAIR

Jessy Cat

@TedShifrin yes!

Ted Shifrin

And, just for your information, @Jessy, I took one graduate course in PDE approximately 40 years ago. I am far from an expert. Even the course I taught that covered Fourier series and PDE was 30 years ago.

Jessy Cat

@TedShifrin okay so I exaggerated. You still know more than me.

Krijn

01:28

"physicists don’t write the sum sign, but remember that one should sum
over indices that repeat twice - once as an upper index and once as lower. This convention is called the Einstein summation, and it also stipulates that if an index appears once, then there is no summation over it, while no index is supposed to appear more than once as an upper index or more than once as a lower index."

Ted Shifrin

Yes, that's true. :) No insult intended.

Krijn

What the hell

Ted Shifrin

Einstein convention has been around for 100 years, @Krijn, although I tend to draw the summation symbol "irregardless."

Krijn

I've heard of it but never looked into it

Semiclassical

Think of it like this. If you wanted to write down the matrix multiplication $ABCD$ explicitly, you'd write $\sum_{klm}A_{ik}B_{kl}C_{lm}D_{mj}$

Ted Shifrin

01:30

The notion that upper and lower indices need to combine and get summed to get something that is well-defined is actually quite powerful.

Danu

@Krijn This is super handy.

Ted Shifrin

No, @Semiclassic, you need one upper one lower for a linear transformation.

Semiclassical

Eh, I'm not worrying about contra/covariant at the moment.

Ted Shifrin

Einstein summation convention demands (appropriately) one upper against one lower.

I am.

Jessy Cat

In any case, I'm getting back to work. TTFN.

Danu

01:31

@TedShifrin Lower them with the metric in Euclidean space and nobody cares. Only once you write Greek letters, it becomes important (duh!)

^the physics way (i'm just joking)

Ted Shifrin

Bull**** @Danu.

Danu

hahahha

So triggered :D

Krijn

Is this just invented to irritate people like me?

Ted Shifrin

waits to be banned from chat

Semiclassical

The thing that's funny to me

Danu

01:31

@TedShifrin Oh, physics and math

Ted Shifrin

No, @Krijn, it actually works out well with Hermann Weyl's theorems on invariant theory, etc.

Danu

What a strange marriage

But what beautiful children.

Ted Shifrin

I'm sure Trump will do away with that, too.

Danu

Don't tell me about it :(

Krijn

@TedShifrin O. Well then, proceed

Ted Shifrin

01:32

LOL

Semiclassical

I really don't see much use of Einstein convention in quantum mechanics

Hiro

Just a question, does an altitude bisect a side in a triangle?

Danu

It's essential only in relativistic theories

Semiclassical

But in quantum mechanics, you have the whole bra-ket thing.

Ted Shifrin

Noooo @Hiro, very rarely.

Hiro

01:33

oh...

Ted Shifrin

Only when you're dropping an altitude from a vertex with two sides of the same length coming in to it.

Hiro

Well that makes things 180 degrees more complex

Ted Shifrin

i.e., an isosceles triangle.

Semiclassical

You could get rid of the bra-ket thing and just have stuff like $\phi_i=|i\rangle$ and $\phi^i=\langle i|$

Hiro

oh it bisects an isosceles triangle?

Krijn

01:33

But then, how do you differentiate between the sum of $a_ix^i$ and the actual $a_ix^i$

Semiclassical

not sure which order I like better, w/e

Ted Shifrin

Draw the picture and prove it, @Hiro.

@Krijn: It gets particularly troublesome in differential geometry when coordinates have upper indices and you want to take powers of those variables.

Semiclassical

In general, if you're doing Einstein notation you don't care about individual a_i x^i

Ted Shifrin

If you do, you have to say "unsummed" IN WORDS.

Semiclassical

Right.

Ted Shifrin

01:35

But that does NOT mean exponent when you write that $i$. It means $i$th variable.

Semiclassical

Yeah.

Ted Shifrin

Yeah.

Semiclassical

Hence why it's not convenient if you want any kind of multiplication.

Krijn

Yeah, I think that's actually quite nice

Semiclassical

It's mostly handy if you're doing contraction of tensors.

Ted Shifrin

01:36

So dot product shows up as pairing a covector with a vector, and the notation works perfectly, @Krijn.

If you want a dot product of two vectors, you need the metric tensor in there: $g_{ij}x^iy^j$.

Semiclassical

There's also the connection to diagrammatic notation, if you're Roger Penrose :)

Ted Shifrin

@Krijn: There are also unitary/hermitian variants, where you put a bar over the second index and it all works out swimmingly.

Krijn

@TedShifrin Ah, yeah, we use the bar for practically nothing so no chance of confusion there

Ted Shifrin

Bar for complex conjugation.

Totally consistent.

Hermitian inner products are, after all, sesquelinear.

Krijn

I do believe it all works out pretty nicely in the end

It's just not for my taste, I think

Danu

01:42

@Ted I found Gilmore's book "Lie Groups, Lie Algebras and Some of Their Applications".

This seems to be pretty nice

Ted Shifrin

I know nothing, @Danu.

Danu

Starts by defining a set, a group, etc :D

Ends with Dynkin diagrams

Claims to have been written based on notes for a course about Lie algebras and their applications in physics

Looks perfect for physics students (that want to learn some math)

Ted Shifrin

Well, give it a shot.

Danu

I'm just sending them an email with stuff I recommend

Hiro

@TedShifrin Let ABC be a triangle, let a,b,c be the lengths of its sides opposite to A,B,C respectively, and let $h_A$, $h_B$, and $h_C$ be the lengths of the altitudes from A, B, and C. Suppose that $\sqrt{a+h_B} + \sqrt{b+h_C} + \sqrt{c+h_A} = \sqrt{a+h_C} + \sqrt{b+h_A} + \sqrt{c+h_B}$. Show that $(a+h_B)(b+h_C)(c+h_A)=(a+h_C)(b+h_A)(c+h_B)$

Danu

01:45

I guess nobody will even look at it

but it's worth a shot to try 'n' inspire them

Ted Shifrin

Ugh @Hiro.

Semiclassical

ew

Ted Shifrin

Even though I'm geometric, things like this make me go "ick."

Hiro

lol

@TedShifrin it's not that bad xD

Danu

Plane geometry is the worst geometry

Ted Shifrin

01:47

What the hell is the meaning of something like $\sqrt{a+h_B}$?

Danu

There, I said it

Ted Shifrin

There can be beautiful things, @Danu.

DHMO

@TedShifrin is the point that we can't?

Danu

can, maybe. Nobody showed me much cool stuff.

Ted Shifrin

We can't what, @DHMO?

I can sometime, @Danu.

Semiclassical

01:47

I think that's not necessarily as complicated as it seems, though. If you write it as $\alpha_1+\alpha_2+\alpha_3=\beta_1+\beta_2+\beta_3$, then the implication is $\alpha_1^2\alpha_2^2 \alpha_3^2=\beta_1^2\beta_2^2\beta_3^2$

Ted Shifrin

Yes, @Semiclassic. I saw that much.

Danu

@TedShifrin Would love to hear about it.

Ted Shifrin

That doesn't feel geometric to me at all.

Semiclassical

and, ignoring the squares, those terms show up if you cube both sides

Ted Shifrin

Huh?

Krijn

01:48

Gauss thought his 17-gon was pretty spectacular and he's not a bad mathematician, I think

Hiro

hmm?

Ted Shifrin

"Ignoring the squares"?

Hiro

^

Ted Shifrin

I am not arguing at all, @Krijn.

Krijn

No no

Semiclassical

01:49

$\alpha_1\alpha_2\alpha_3=\beta_1\beta_2\beta_3$

Krijn

@Danu was

Ted Shifrin

Yeah, but what about all the rest of the junk, @Semiclassic. Surely there's some geometry in there somewhere?

Hiro

from where did you come up with that expression? @Semiclassical

Semiclassical

Oh, sure. But my point is that, if you cube both sides, then that relation can only be true if there's other such relations.

Ted Shifrin

@Danu: In both my books (where it's relevant) I have the students do vector proofs that the altitudes, perpendicular bisectors, and angle bisectors intersect, respectively, in three points. Then you want to show that those three points are collinear, and — in the appropriate order — are in a 2:1 ratio. I think that's all gorgeous.

Semiclassical

01:51

By labelling each of those square roots as $\alpha_k$ or $\beta_k$

Just to get to less tedious algebra.

Ted Shifrin

@Semiclassic: True enough.

DHMO

@TedShifrin We can't prove $\not\exists(a,b)\in\Bbb N^2: a+a=b+b+1$

Ted Shifrin

LOL @DHMO.

DHMO

@TedShifrin can we?

Ted Shifrin

Well, you can if mod 2 is well-defined. :)

Danu

01:52

@TedShifrin Because of the nice pictures, or is there a better reason to like it much? :P

Ted Shifrin

But that begs the question (which we all take for granted).

DHMO

@TedShifrin no, I don't [take for granted].

Hiro

if $\alpha_1+\alpha_2+\alpha_3=\beta_1+\beta_2+\beta_3$, how does that imply $\alpha_1^2\alpha_2^2 \alpha_3^2=\beta_1^2\beta_2^2\beta_3^2$

Krijn

@Danu The geometric aspect of Galois theory is pretty as well

DHMO

@TedShifrin that is essentially the statement itself

Hiro

01:53

^^ @Semiclassical

Semiclassical

You've also got $\alpha_1^2+\alpha_2^2+\alpha_3^2=\beta_1^2+\beta_2^2+\beta_3^2$

Danu

@Krijn I don't know any Galois theory :(

Ted Shifrin

@Danu: I don't have physical applications. Although, now that you bring it up, there's another point which is fascinating. What's the point that minimizes $Ax + Bx + Cx$ for vertices $A,B,C$ and point $x$?

Semiclassical

I didn't say I knew, I was just reformulating it

Danu

@TedShifrin I guess I'm supposed to put $x$ somewhere inside the triangle?

Semiclassical

01:54

@TedShifrin I don't quite get that expression

Ted Shifrin

@Krijn: Unless you mean arithmetic algebraic geometry, the geometry is reaching.

@Danu: Maybe.

But sure ...

for starters.

Hiro

yeah $\alpha_1^2+\alpha_2^2+\alpha_3^2=\beta_1^2+\beta_2^2+\beta_3^2$ makes sense, but what do you have to do to get to $\alpha_1^2\alpha_2^2 \alpha_3^2=\beta_1^2\beta_2^2\beta_3^2$ @Semiclassical

DHMO

@Null Objection: you essentially labelled yourself and the pope with the same label. That does not make you the pope.

Semiclassical

Should that be $|AX|$, for instance?

Danu

$Ax$ meaning... Inner product?

Krijn

01:55

@TedShifrin Que?

Ted Shifrin

@Semiclassic: Of course. I was being lazy.

Danu

Ohhh

I see

Ted Shifrin

No, no. What Semiclassic said.

Danu

I forgot the plane geometry notation :P

Semiclassical

That'd be the Fermat point, if memory serves

Ted Shifrin

01:55

I tend to prefer $\|\overrightarrow{AX}\|$, but I was lazy.

Hush @Semiclassic.

Semiclassical

My prof asked me a similar question, but with 4 vertices on a square..

Fiine.

Danu

Spoiler :D :P

Semiclassical

It made me think for a bit :)

Danu

So the point is how to construct it I guess

Ted Shifrin

Very not unique for a square.

Semiclassical

01:56

And it actually had a connection to the physics we were working on, amusingly enough.

Hiro

@TedShifrin how can you rearrange $\alpha_1^2+\alpha_2^2+\alpha_3^2=\beta_1^2+\beta_2^2+\beta_3^2$ to get $\alpha_1^2\alpha_2^2 \alpha_3^2=\beta_1^2\beta_2^2\beta_3^2$?

Ted Shifrin

Right, @Danu ... and it has a physical interpretation, unlike the others.

@Hiro: SOMEWHERE you have to use some geometry :P

Hiro

@TedShifrin like what kind of geometry are you talking about lol

Danu

@TedShifrin I might think about it some other time---I have to admit that those construction problems can be kind of fun. Do you know the game Euclid?

Ted Shifrin

lengths and altitudes, @Hiro.

No @Danu

Danu

01:58

You have to do constructions in plane geometry, progressively harder ones.

Semiclassical

I'm not sure how much of this is really geometry, though.

I think it's mostly just implications from algebra, and that's pretty boring.

Ted Shifrin

I did say ugh^3 and ick a few times, @Semiclassic.

Danu

As you show you're able to do some constructions, you are allowed to use them (via a "shortcut button") in the next levels. euclidthegame.com/Tutorial

Semiclassical

Yeah.

Danu

It's super cool

Ted Shifrin

01:58

I bet Pythagoras comes in ...

Danu

The last few levels took me a while :D

Semiclassical

probably.

Ted Shifrin

I won't even go there, @Danu.

Danu

Of course, the real challenge is doing the levels with minimal number of lines ;D But I'm not hardcore enough

Semiclassical

I mean, you can write down the area of the triangle in three different ways

Danu

01:59

D'aww Ted. It's so much fun!

DHMO

For $a,b\in\Bbb N$, prove that exactly one of the following is true:
$\exists n\in\Bbb N:a+n=b$
$a=b$
$\exists n\in\Bbb N:b+n=a$

Ted Shifrin

I'm not so much a game-player, @Danu, other than bridge.

And cribbage.

Semiclassical

and that imposes some further relations on the altitudes and lengths.

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$Mathematics$ Mathematics