Mathematics - 2017-07-27 (page 4 of 4)

despite me oddly preferring it over the first (guy skips alot of steps or uses only one example)

first ones authors are Daniel Norman and Dan wolczuk

Strang and I are good friends, and I stole some ideas from him in writing my book.

OK, I don't know the other book.

I found the following in Besse's book (paragraph 9.53)

Actually, one of the things I'm proudest of in my book, @Faust, no one ever teaches, and that's the computer graphics discussion in the last chapter.

reading stranges book after taking a linear algerbra class it seems not bad

11:01 PM

I probably can't answer, @Danu, but go on.

but if i was trying to learn from it i would be super lost

I need to use this last identity, but I don't really understand how to derive it.

he only uses one example and solves it 3 diffrent ways alot of times

@TedShifrin Is linear algebra related to it? :P

11:02 PM

More examples = more better!

Oh, @Danu. That's the usual principal bundle definition of curvature (see Kobayashi-Nomizu, for example) if $M$ is the frame bundle over $B$.

Can you elaborate?

(precise reference to KN would also be great!)

@TedShifrin numerical analysis?

What I remember is that if we have a connection one-form $A$ then the curvature is $F=dA+[A,A]$

@TedShifrin Any chance you have old problems sets from undergraduate classes laying around?

11:06 PM

@Danu :O Since when were you a moderator?

@Danu: That's mixing apples and oranges.

@TedShifrin Right.

So can you elaborate? :D

@SimplyBeautifulArt Years, literally.

I'm not a moderator on Mathematics.

Oh lol

I'm a moderator on History of Science and Mathematics

Look on p. 76-78 of KN, @Danu.

11:07 PM

@Danu That's cool

Vol 1 or 2, Ted?

1

@Faust: Like for what classes?

In 1, I'm just seeing the structure equation as I just wrote it down explained

hello, i have a question about a topology video i watched.

The formula you asked about is on p. 78, @Danu.

11:09 PM

Ah, do you mean corollary 5.3?

it said that a donut and a coffee mug are topologically the same, but i don't get that for two reasons. first, it said to think of deforming the donut as if it was made of clay, without cutting or pasting, and by that logic, a donut and a sphere are topologically equivalent - just smoosh the donut in and reform.

I put the book away already.

No, @heather, you can't get rid of the hole in the donut.

I quit life

@TedShifrin so clay is just a poor analogy?

the other thought is that the coffee mug has a hole for the handle, and a "hole" that is the emptiness in the middle of the coffee.

@LucasHenrique Same tbh

11:10 PM

I was supposed to buy the abstract algebra book. The. Abstract. Algebra. Book.

no, the place the coffee goes is outside of the surface, but the handle is the hole of the doughnut.

Did you buy linear algebra? @Lucas

I wondered, @Lucas. You probably shouldn't be wasting money on books, anyhow :P

Yes.

@TedShifrin Welp, I want to be a mathematician so probably I'll waste a lot of money on books.

Lucas, send me an email.

But my point was that if you were going to get linear algebra, you could have done that in Portuguese :P

11:13 PM

Once you're in college you'll have a library available so you won't have to worry too much about purchasing them

which is more holes than a donut, but i guess you're not punching through the bottom of the mug, so it's technically only one hole...

better to make things out of thing rubber than out of clay, @heather.

but then how do you get from a donut to a coffee mug?

@TedShifrin Anything interesting i have two 4 year gaps of no math in my short life and only having 4 months back at it this time around, well to be honest i have probally forgotten more than i ever learned.
the most heavy things i need to review still are linear algebra and multivariable calculus but id love some stuff on topology as well. the more problems i do the faster i seem to remember how everything works

problem is that most of my homework sets refer to some textbook, @Faust, either my own or someone else's.

11:15 PM

ah fair enough =)

@heather: grab on to the rubber donut by one "side" and push the other side around to make the coffee cup without the handle part.

@Faust: I can send you lots of exams, I suppose.

perhaps you could recommend a good multivarable calculus textbook i refuse to open mine ever again anyway :p

@TedShifrin i'm having trouble with what you mean, i'm sorry

@Faust: For that I obviously prefer my own :P

@TedShifrin yes, I bought it in Portuguese

11:17 PM

@heather: I don't know how to do this without pictures. Are you grabbing the rubber doughnut with your hand around the right part and your fingers going through the hole?

@Lucas: Well, that's good, at least. Email me re the other book.

@TedShifrin yeah, grasping it like a suitcase handle.

Doing this at this exact moment. Thank you so much, professor.

Ah, DogAteMy to the rescue.

Gif:

11:19 PM

Thanks!

Poifect.

@AkivaWeinberger ah...lightbulb.

resigns and gives the reigns to DogAteMy

@Ted, I sent you an email.

11:20 PM

OK, @Lucas.

i have another question, in the same vein - take a cylinder. that's topologically equivalent to a sphere, if i understand right. then stretch it, begin to curve it, until it is infinitesimally close to the starting point. that's still as I understand it equivalent to a sphere. if i then connect the two ends, that becomes a torus. why is that different?

@TedShifrin seems my libary has it for some reason the used ones are alot more then the new ones on amazon O.o

A cylinder with top and bottom? Like a can of soup.

@TedShifrin Seems like a reigny day

@Faust: Prices are absurd. I cannot justify them.

11:21 PM

@heather Does your cylinder include the interior? In any case: Is the reverse transformation continuous?

Or would you have to cut it to reverse it?

@AkivaWeinberger I always see this gif and I know it's related to math. Topology, differential "things" maybe?

I still don't quite understand what's going on @TedShifrin. Everything here is about curvature of principal bundles. I also don't quite understand what $R^\nabla$ is supposed to be in the picture I linked.

@LucasHenrique Topology. Showing that a donut is homeomorphic to a coffee cup.

@Akiva, I'm picturing a solid can of soup, and yeah, you'd have to cut it to reverse it, once you fully connect it. but if you get it infinitesimally close, no, you wouldn't have to cut it.

I understand that the structure equation gives us $\mathcal V[X,Y]=-F(X,Y)$ for horizontal vectors on a principal bundle

11:22 PM

your not supposed to cut

@heather In addition, the transformation isn't bijective since the two circular faces of the can end up overlapping

It's just the curvature 2-form as a Lie-algebra valued form, @Danu.

I'm pretty sure $R^\nabla$ is supposed to be the curvature on some vector bundle, not a principal one.

@heather What does "infinitesimally close" mean? If there's still space between the two circular faces, it's not a torus

@Akiva, okay. i'm sorry, i'm not thinking about this quite as mathematically, just more playing around with it.

11:23 PM

Doesn't matter. You can do vector bundle or associated frame bundle. No difference.

You still get a $\mathfrak{gl}(k)$-valued form for curvature.

@TedShifrin ill try the rental from the library and if i like it then ill drop the 200 bones on it.

@AkivaWeinberger this kind of thing makes both happy, ("good") anxious, ("bad") anxious and scared of uni-lelel mathematics

i just need to keep reading about it.

hi

@Faust, don't waste money ... :) Also, use the videos to help.

11:25 PM

@Meow hey hey hey look who it is!

@Meow!

Also, what is $\pi_* $ doing there then? $R^\nabla$ does not seem to be a curvature form on the total space, but rather on the base @Ted.

hi @TedShifrin @Daminark

how was europe ted

@Meow! Haven't seen you in a while, how's a going?

im ok most of the things ive been doing have to do with software

11:25 PM

@Danu: It doesn't matter. Curvature descends to the base anyhow.

how about you?

OK, I'm trying to do 5 things at once, and I cannot.

It matters to my level of understanding what's going on!

I'll lay off

@MeowMix cool, what sorts of things?

operating system development

11:26 PM

I can ask a PhD student, since this is supposed to be easy I guess

game disassembly

wow

and since its summer my sleep schedule is fucked :]

Welcome to the club

i wake up at 2, have breakfast and go back to sleep until about now when i get dinner and wake up

11:27 PM

@TedShifrin perhaps but a good book is a good book and i didn't waste those 8 years i wasnt in school =) but ill check out the video lectures.

@heather how are your studies going?

Anyway im going back to re-learning math you guys have a good day

@MeowMix pretty good, reading tons about quantum computing, working on an experiment I hope to conduct. Dabbling a bit in optics and math as it comes up in my reading.

haha i watched a video on quantum computing today

i was like "what the fuck" and got back to my silly classical computng nonsense

=/

what video?

11:30 PM

im not sure, i think it was by IBM

i found michael nielsen's videos pretty good

if you want, i have an introduction with some math written up (if you don't that's fine too)

i swear if im going to have to write quantum assembly my brain is going to be fucked

feel free to drop that link, ill watch it when i get a chance

michael nielsen videos: youtube.com/watch?v=X2q1PuI2RFI&list=PL1826E60FD05B44E4

my current draft (for the worldbuilding blog): medium.com/p/db86d1f1665a/edit

alright thanks, putting into firefox pocket

and if you want, there's a great little simulator, quirk, that you can mess around with.

11:37 PM

anyways just wanted to stop bye

hmm i definitely will, want to mess with it before the inevitable takeover of quantum computers

well, not really an inevitable takeover

you're not going to use quantum computing to check your email; no efficiency gain

really, considering the currently discovered algorithms, it's going to have the biggest impacts on cryptography and communications

i work with servers

so yes, my rsa shit is gonna be inevitably cracked and ill be sad :(

maybe searching, though grover's algorithm doesn't have nearly as efficient gain as shor's algorithm

@MeowMix well, but there'll be new algorithms to protect things.

i hope

dont want quantum strangers ssh'ing without permission

this seems interesting to mess with

im going to see what i can come up with

i'd also check out qasm

quantum assembly

11:41 PM

thats a thing?????

everyone throw out your x86 computers, the future is now

you may also wish to pop into the hbar and talk with Daniel Sank (a quantum computing researcher at Martinis Lab) and Bernardo Meurer (college student interested in quantum computing and computers)

heh i dont think i have the smarts

@MeowMix yup, if you use ibm quantum experience you can download your programs as qasm

id probably fit more in some software oriented chat

@MeowMix i knew nothing a couple months ago.