Mathematics - 2019-06-07 (page 3 of 3)

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8:09 PM

@Daminark Got it ; D

Érico Melo Silva

8:22 PM

@RyanUnger oh cool

isn’t he ancient

8:34 PM

@ÉricoMeloSilva 83

but we have to show up for at least one lecture :P

Be aware: Training a single AI model can emit as much carbon as 5 cars in their lifetimes

Érico Melo Silva

ya no question i’ll show

9:25 PM

Problem: Explain why the action of the group of rigid motions of a cube on the set of three pairs of opposite faces is not faithful. Find the kernel of this action.

Question: What exactly does the group look like; and what exactly is the set that this group is acting on? How is it acting?

9:58 PM

Im reading this book on dynamical systems

and I came across the following theorem

user image

I get if I take the derivative why it will be of the form $X' = AX$

but why is this necessarily the general solution? does this theorem have name of some sort?

@StanShunpike this only works in 2D

generally the solution to $\dot{\vec v} = A{\vec v}$ is $\vec v = e^{tA} \vec v_0$

where $e^-$ is the matrix exponential

@LeakyNun Ok thanks! I was worried about that cuz my book seems to do mostly 2D examples. Do you have a recommended textbook for introducing n-dimensional dynamical systems?

or like

n dimesnional vectors

cuz the matrices they consider are 2D

sorry I don't know which book to recommend

ok thanks

11:08 PM

@StanShunpike: Look at Hirsch/Smale. You can also see this discussed in my videos (one of the last few lectures). You get a general solution of that form (uniquely) whenever $A$ has distinct eigenvalues. With repeated eigenvalues, you need to deal with Jordan normal form. (That's done in depth in Hirsch/Smale, not in my lectures.) Oh, and hi :)

@TedShifrin my USA visa got approved!

Congrats!

Oh, congrats, Leaky. I guess it's happening. :)

You're not from a s***hole country, so I figured you wouldn't have a problem.

Oh wait, we're one of those now.

@StanShunpike: Actually, I could have made a slightly stronger statement: You don't need distinct eigenvalues, just a diagonalizable matrix (i.e., a basis of eigenvectors).

That's exciting, Leaky!

Hi @Rithaniel

11:19 PM

Heya Ted. How's it going?

Life's a bit crazy at the moment, but I'm relatively fine, thanks.

How you?

I've got a job at the university over the summer. Just helping the administrative people keep things organized. Other than that, being lazy and trying to figure out what I want/need to do for grad school.

Aha.

Hello

Heya nerd Demonark

11:26 PM

Sup Daminark.

How's everything going?

Wait a nerd? Where?

When are you moving to Madison, Demonark?

August 15th

Oh, so you're a bum until then?

Yup, in fact just today I finished my last final exam

11:28 PM

@TedShifrin hey ted!

heya @Stan

Don't get too used to bummery, Demonark.

@TedShifrin got a primary advisor for my masters thesis and a co-advisor. I'm going to be doing it in medical applications of robotics. I've got a working thesis topic I'm really excited about. my program doesn't start til fall but i'm gonna be working this summer on it

but i've got to educate myself about ODEs, PDEs, and dynamical systems

Haha, I'll try to keep the math up in the meantime for sure

the dynamical systems text im using is working with mostly 2D matrices to start with and I can't tell if thats a good thing or bad thing

Some mix of bringing myself up to a level where I can definitely knock out at least one qual and just fun math

11:32 PM

All good stuff for you to be learning.

That's exciting, @Stan.

@TedShifrin Yeah! Since I've had winter and spring free, ive been studying math everyday

to give myself the tools i'll need to succeed

basically taking areas i suck at

and raising my competency level

You staying at UC or moving elsewhere?

still at UC

Oh for a second I thought you were talking to me

That wouldn't make sense, Demonark.

11:34 PM

My "fun math" of choice these last couple months has consisted of figuring out the cayley table for the multiplication operation on finite fields.

You didn't know that Stan has been at UC, too?

Something is rotten in the state of @Daminark

@Rithaniel: You mean an ad hoc calculation each time you run into a (small) finite field?

just kidding

i just saw hamlet

it was so good

Well, technically speaking I've not run into any finite fields. I'm just calculating them for the sake of calculating them.

11:35 PM

@TedShifrin what have u been up to?

Wait I'm not exactly sure who you are off the top of my head :(

@Daminark me?

Probably retiring completely from teaching, @Stan. My two years teaching for Art of Problem Solving have been fun but frustrating, and I'm not going on.

@Daminark I'm the guy who drives the Knight Bus

Though, yeah, I suppose "ad hoc" is an accurate descriptor

11:36 PM

@TedShifrin What made it frustrating?

I remember you expressing this before

Oic

Oh yeah, I knew I recognized that name from somewhere!

It's been forever since I read Harry Potter, though.

Students hardly did any work at all outside of the 1 3/4-hour classes. I wrote all sorts of extra stuff, and only 2 of the 5 students even looked at those. I had told them to write up a few problems and let me criticize their writing skills (the only AoPS homework for my class was on-line). I think I got about 4 problems turned in (by two students) all year.

Plus I did a certain amount of stuff that was different from their pre-designed course, because I wanted to do some proofs or enhance understanding a bit better. But anyway, meh.

Feeling underappreciated can suck. I suppose that's the main source of stress for teachers, though.

@TedShifrin I experienced strong disinterest from the students I was teaching. Didn't matter how much I tried to relate to them or make the material exciting

they just didn't care

11:40 PM

Well, some teachers don't really care much about teaching :)

I was fortunate that most of the courses I taught at UGA had students who really wanted to be in them. But when I taught a core course or a required major course, there were people I couldn't motivate.

Thinking about it, actually, seeing a professor who is still is enthusiastic about teaching even though they've been doing it for years is surprising.

Hello all. I was just wondering, is the regular value theorem an 'iff'? I mean this in the following sense: If I take f\in R[x_1,\dots,x_n] and take f^{-1}(c) where c is a regular value, then I know that this is a smooth submanifold of R^n.

But in general, if c isn't a regular value, then f^{-1}(c) is surely still a topological submanifold, but it may not be smooth. Can it ever be the case that f^{-1}(c) is still a smooth manifold, yet c is not regular?

@Rithaniel, if you watch a few minutes of my YouTube videos, I think you can tell I still cared and engaged with the students.

@F.White: Yes, if you mean the underlying set. Just consider $f(x)=x^2$, $f\colon\Bbb R\to\Bbb R$. You can jazz that up easily.

@TedShifrin hi

hi @Adeek

Oh, @F.White, obviously, I want you to take $c=0$.

11:42 PM

I'll have to check them out. Do you have any lectures on, say, non-associative magmas?

@TedShifrin got 18k scholarship :D

@TedShifrin your enthusiasm for math is great. i remember before talking / meeting you how much i felt like math mattered when i watched your lectures

My partner though she will take care of the money. I would spend it on books.

@TedShifrin only one year left and you'll have a... different... president

@TedShifrin its a shame thats not always a priority with schools

11:43 PM

@StanShunpike TED Is the BEST

my high school physics teacher is retiring

Good for you, @Adeek. Congrats.

@Leaky: That is far from clear.

and its over school politics

and he's unbelievable

did his PhD with Peter Higgs

Érico Melo Silva

@TedShifrin lol rip

@TedShifrin This can happen outside of the zero-dimensional case though? It seems these sort of examples always use double-points etc, and these surely can't be smooth

11:43 PM

That is so sad for the students who won't get him, @Stan.

@TedShifrin I agree. He quit because the admin had terrible hiring practices and apparently didn't consider a candidate he thought more than merited their attention. I think he then felt this wasn't a good use of his time.

teaching i mean

I hope that U.S gets an different president.

and yeah its a real loss. to this day I credit him with getting me started in physics and helping me love it

@F.White: Yes, we can jazz that up into as many dimensions as you want. Take $f\colon\Bbb R^n\to\Bbb R$ with $0$ a regular value, and consider the function $f^2$. You can also take sums of squares (a failing of $\Bbb R$ that does not occur with $\Bbb C$).

@Stan: I think it's agreed that exceptional teachers have a huge impact. Sadly, so do the exceptionally bad ones, in a bad way.

I'll think about this, thanks

11:46 PM

howdy @Eric

Érico Melo Silva

hi hi

@Rithaniel Hell no. I wish I could have rounded up students to video the undergrad diff geo class my last semester, but they were all worn out.

I actually was thinking about taking either algebraic topology or differential geometry in my last semester.

So one thing i'm confused about with dynamical systems is the idea of like a phase portrait where you have these plots with all the arrows. specifically, I'm confused about why people find these so valuable. isn't it very limiting to only have two dimensions?

Undergrad diff geo or grad?

11:49 PM

What do you do for multiple dimensions?

I'm a firm believer one should take a good undergrad course before the fancy grad stuff.

It would have been independent study. So just one professor and me.

@Stan: Yes, those are good for providing intuition in 2D. And we use 2D intuition to build intuition for higher dimensions. As I kept complaining, my 3D blackboard was still on back order.

Oh, but you didn't do it, @Rithaniel?

Though, as it is now, I'm likely going to be taking three grad level courses. General probability, abstract algebra I at grad level (already have done that one at undergrad level) and matrix analysis.

@Rithaniel make sure you have the time.

Grad classes aren't like undergraduate one's

11:52 PM

Probability assumes a graduate real analysis course (measure theory, Lebesgue integrals)?

@TedShifrin hahahahahahhaa what a great line

@Adeek: You need to edit.

@Stan: It appears in my lectures a few times, I think :)

Done

And I need to complain about your apostrophe use, just like I do about Balarka's :P @Adeek

I will. If there is anything about me, it's that I work quickly. Also, yes, on the measure theory. Though that course was very relaxed. We didn't even have a final exam in there. So I should probably look into some review.

11:53 PM

@TedShifrin Yeah that would be nice! I actually never took English grammar ever my english comes from reading books.

@TedShifrin so learning the 2D ones won't be a waste of time? I was worried that like, it will be like for the cross product, where it will be hard to transfer

@Rithaniel: OK, I still would vote for going through something like my undergrad diff geo notes (freely available). It will give you a better understanding of some stuff you don't use enough, plus give you intuition when you do grad work in geometry/topology.

when i learned the cross product, the higher dimensional analog really wasn't similar

@TedShifrin The way I aced the essay part of SAT was memorizing like 100 essays on different topics

ever since i've been wary of stuff that's not generalized immediately

11:54 PM

when I took it few years ago.

@Stan: Nah, it's fine to start with. But look at Hirsch/Smale. It's a fabulous book.

@Stan: Cross product is really special because we should be doing exterior algebra (and differential forms) instead. :P

No problem generalizing $\Lambda^2 V$ to higher dimensions :)

Will do. That should give me some stuff to read over the weekend.

Also, I was looking into taking a math GRE but couldn't find a place which was holding one.

LOL, it might take a few weekends at the very least :P

I will get a grammar book.

Ah, I see you have homework assignments laid out, too.

11:57 PM

@TedShifrin i finally worked through how the wedge product works the other day

You mean subject GRE during the summer? Yeah, they don't do that. But you should be able to find one for early fall and then later fall. Make sure you do the early one.

@TedShifrin but now that i've done that, i'm more confused that ever what a torque is

i of course know the simple definition

Torque is derivative of angular momentum :)

but i just don't get the intuition with wedge products

You need to work with them and then intuition magically happens.

@Rithaniel: I do? I mean, I can send you homeworks that I assigned. Lots of good exercises in the notes, if that's what you meant.

11:59 PM

This is the website I've found myself on: alpha.math.uga.edu/~shifrin/MATH4250

Ohhh ... I'm amazed that's still there. You can find the link to the latest version of the notes in my profile on here. I did some revisions.

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