Mathematics - 2014-11-30 (page 1 of 4)

teadawg1337

00:45

Holy cow, writing proofs like that is exhausting...

Never before have I gotten a headache while doing math. This problem has been bugging me for almost two weeks now, and I've finally solved it! Now there's a more challenging question to answer: what should I do next???

teadawg1337

01:07

Sorry guys, I'm not very good with introductions...

Dillon

Hey everyone ! :). If anyone has some knowledge in equilibrium stability analysis, please have a look at this question for me: http://math.stackexchange.com/questions/1044269/sirs-equilibria-stability-analysis

I really appreciate it! :). I created a chatroom for discussion as well..
I'm trying to teach myself some SIR models and would really appreciate all the help I can possibly get.

Committing to a challenge

I don't personally sorry. But hopefully after American dinner people can come and help you.

teadawg1337

I also can't help with that topic, sorry

Dillon

Thank you anyways! :). I appreciate you guys having a look at it :)

teadawg1337

01:22

@Committingtoachallenge Wait, it's already 7:23 in the evening??? I really need to put a clock in my room...

Dillon

Hehe! It's 3:23 AM here :P

Committing to a challenge

It is 11:25AM here

My american clocks are at 5:26PM and 8:26PM

PST and Eastern

teadawg1337

I live in Central

Dillon

I live in South Africa :)

Committing to a challenge

I live in Australia

teadawg1337

01:32

We may live far apart, but we all share a passion for mathematics!

Committing to a challenge

:)

teadawg1337

I personally feel that calculus in particular is subject to unwarranted hatred

Allow me to clarify: it's a beautiful subject, and it's notoriously "challenging" and hated by many people

Dillon

A lot of people tend to find calculus intimidating and hence they have a very negative view towards it ... Especially multivariable vector calculus. But as for the topics that we've had so far, the one students hate the most here, is definitely Real Analysis (which is my favorite)

teadawg1337

Multivariable calculus is especially beautiful, and Real Analysis is a blast as well

Committing to a challenge

I personally dislike the 'Engineering' approach they teach calculus, I wouldn't call that real math, and that may be part of the problem

That being said, I certainly don't dislike calculus

teadawg1337

01:53

I LOOOOOOOOOOOVE calculus. In fact, I finished teaching myself three college semesters worth of it a few weeks ago. It took me a month, including vector calc

It's just....... So..... Perfect. BETTER than perfect, really.

Calculus is perfect, not my talent for mathematics

Wow, my modesty goes down the drain when I'm super tired

teadawg1337

02:09

I really need more challenging problems like this one...

Karl Kronenfeld

02:30

@anon I was reading your comment on symmetric/alternating groups, and I can't seem to make Sym a functor.

Well, I could do something trivial like having Sym(f) be nontrivial iff f is a bijection. But then the question you're asking becomes meaningless in a categorical context imo.

Committing to a challenge

@teadawg1337 My Calculus was very surface level, so I am relearning it from Zorich right now.

teadawg1337

@Committingtoachallenge I've heard that Zorich is a good read, but I'm a bit tight on money right now. I found a copy of Apostol's Mathematical Analysis at a garage sale for a dollar, I practically threw my money and ran home xD

Mike Miller

02:50

@Karl Why isn't it a functor? If $1$ and $2$ map to $x$ and $y$ send $(12)$ to $(xy)$.

Karl Kronenfeld

suppose x=y, 3 mapsto z, 4 mapsto w: (13)(24) mapsto (xz)(xw)

Mike Miller

don't see the problem

sorry if I'm being dense

Karl Kronenfeld

then the image permutation is not well-defined if z != w

Mike Miller

sure it is, just like (13)(23) is well-defined (as the product of these)

Karl Kronenfeld

unless we're doing the usual multiplication, which I did not even think of tbh

Mike Miller

02:56

thay's what I was thinking of

Karl Kronenfeld

(realizes it now)

Mike Miller

not really sure if this is an interesting functor but it does seem to be one

Karl Kronenfeld

do we get the identity permutation mapping to the identity in a satisfying way

yeah, we do.

Mike Miller

a bit worried it's a homomorphism

Committing to a challenge

@teadawg1337 You can reach Zorich I&II as the first two results of 'zorich pdf' on google. So you can use them alongside apostols

Karl Kronenfeld

03:01

one other potential problem may involve the (insufficient) faithfulness of representing a permutation using transpositions.

Mike Miller

one can do this canonically enough

so wait, what? k consider the map {1,2,3} to {1,2} by 1 > 1, 2,3 \to 2. (123) maps to what, precisely? (122)=??

should this be (12)(13)\to (12)(12)?

in which case this is all awful.

teadawg1337

@Committingtoachallenge Oooh, I will! Tomorrow, though. I can't keep my eyes open anymore, that proof really took its toll on me

Karl Kronenfeld

yeah, I was expecting it to be given by breaking the permutation into transpositions, and applying f to the individual transpositions

Mike Miller

this can still be done canonically but it's not clear that it's well-defined without using a canonical decomposition, and I've lost the will to go on

Alexander Gruber

what are you guys talking about

Karl Kronenfeld

03:07

indeed you don't want to have to break it down canonically if you want the result to be a homomorphism

trying to make the assignment of a set S to the group Sym(S) a functor @AlexanderGruber

Mike Miller

It's code, @AlexanderGruber; you'll recall we're assassins

Alexander Gruber

can S be infinite?

Karl Kronenfeld

yes

Alexander Gruber

dang

Karl Kronenfeld

at least I am interested in that case

maybe not at first though

do you have anything in mind for the finite case?

Mike Miller

03:10

I've already lost interest in the finite case. Karl is powerful indeed

Karl Kronenfeld

@MikeMiller $\infty$ gets even worse indeed :P

Mike Miller

@Ka

Alexander Gruber

@KarlKronenfeld it's isomorphic to the partial orders as a combinatorial species

Mike Miller

right, not even a cycle decomp like we're using

Alexander Gruber

what part are you stuck on?

Karl Kronenfeld

03:13

well, the functions between sets in our category may not be bijective, and there seem to be potential problems when they're not necessarily injective in particular.

Mike Miller

when they're injective everything is fine however you define it

Karl Kronenfeld

it's possible to induce a function between corresponding sets of transpositions

then, we're stuck on generating a homomorphism from this.

Mike Miller

is there a (small) presentation known for $S_n$ with generators $(xy)$ as $x,y$ vary?

Alexander Gruber

@MikeMiller what do you mean exactly?

generated by transpositions?

Karl Kronenfeld

we can formulaically the describe the relations:

Mike Miller

03:18

Write $S_n$ as a quotient of the free group on $n(n-1)/2$ generators by mapping each to one of the transpositions $xy$. can you give me a really small (hell, I'd like minimal) generating set for the kernel?

Alexander Gruber

that made things a little more confusing for me

i mean you can generate any Sn with (12) (23) (34) etc

but they would need to overlap by that much, else, how would we get things like (1n)

Karl Kronenfeld

gotta run. this stuff doesn't sound fun

Alexander Gruber

take one out at the jth position and we get $S_j\times S_{n-j}$

if you're allowing for bigger transpositions though, like, just permutations of order $2$, then you can find a lot of pairs that will generate the whole group- pretty much all of them

Mike Miller

i don't understand your question.

I am mappng to all transpositions. This will generate the group. the kernel of the map I desdribed is the group of relations between these transpositions. I want to know a minimal generating set for this kernel.

It's not worth thinking much about.

Alexander Gruber

@MikeMiller transpositions like $(ij)$ or like permutations of order $2$?

Mike Miller

03:26

the former

Alexander Gruber

i think that you need for it to be (12), (23), (34), ... , (n-1 n) then (up to conjugacy)

Mike Miller

I really don't understand what you're asking but I think it's not worth the time to figure each other out, since it's frustrating to type on my phone

Sorry

Alexander Gruber

Okay :p no problem

anon

@KarlKronenfeld sorry, I'm more used to thinking of B, the category of finite sets with bijections, than full Set

anyway my question was trivial

Alexander Gruber

@anon i like that category.

Karl Kronenfeld

03:36

at any rate, the transpositions idea doesn't work for the "full" Set, consider any surjection [n]->[n-1]

Mike Miller

works just fine for my favorite surjection [2] \to [1] :)

Kaj Hansen

05:18

Hello everyone!

Committing to a challenge

05:31

Hey @Kaj, how are you?

Kaj Hansen

Hey @Committingtoachallenge!

I'm not too bad. Just relaxing before heading back to campus in the morning.

Random question: how's your differential forms?

Mike Miller

@Kaj Uoure correct that you need invariance for every closd curve.

Kaj Hansen

Thanks @MikeMiller. It's been a few years since I've studied this stuff. I think I'm going to spend my Christmas break re-teaching myself :P

Mike Miller

The construction is essentially: Assume you have a closed 1-form whose integral does not depend on path. Let $f(x)=\int_\gamma \omega where $\gamma$ is a path from 0 to 1. This is well-defined. Now prove $\omega = df$.

Kaj Hansen

haha, having some $\LaTeX$ trouble I see. ;)

I certainly appreciate the insight, and I wouldn't hesitate to up-vote if you posted a response.

Mike Miller

05:44

I'm on my phone. It's difficult to write math. You should use the idea to write a proof - as an exercise for yourself and to help the OP.

Committing to a challenge

@KajHansen None existent xD. What do you mean by differential forms here?

Kaj Hansen

@Committingtoachallenge, Mike and I were just discussing this question: math.stackexchange.com/questions/1044566/…

Committing to a challenge

Yes I wouldn't be able to comment on that, unfortunately I surface learnt all of my Calculus(by my universities Engineering design of the courses)

I am redoing calculus properly in Zorichs Mathematical Analysis I

Mike Miller

Outside of UGA, most people learn differential forms in differential geometry.

Committing to a challenge

I will be doing that as part of my uni degree in a year, so third year semester $2$. I imagine UGA does it much earlier then?

Kaj Hansen

05:49

It's actually really funny because I learned vector calculus without the traditional approach. So whenever I see that sort of thing with nablas, etc. I have to translate into forms. That's definitely something I'll be fixing soon enough though.

@Committingtoachallenge, I took Ted's course as a freshman, and most of the second semester is forms.

Mike Miller

I find calculus interminably boring, so I can't comment on that.

Don't tell my students that, though.

Committing to a challenge

;P

Kaj Hansen

That class was hard as ****** too. I've never worked so hard to pass in my life.

Mike Miller

Good!

Committing to a challenge

@KajHansen Oh damn, that would probably be motivational though

Mike Miller

05:51

Well, bad. They should have kept working you just as hard.

Kaj Hansen

I suspect a lot of it was getting out of my terrible high school work ethic, but that experience is also what convinced me to go into mathematics.

Mike Miller

Tell Ted that. Give him some joy before retiring!

Kaj Hansen

I'm fairly sure I wrote something to that effect in his course evaluation. Tbh, he's one of the few professors I've actually bothered to fill one out for, along with a couple other professors in our math dept.

Mike Miller

There's not really a good reason not to. If you like a professor, it's good for their future promotion/hiring/stuff. If you dont, the comments might help them improve their future teaching.

Kaj Hansen

That's true; I can't argue there.

Committing to a challenge

06:06

My university offers prizes for filling it out

All of my texts are stuck on large exercise lists and it is demotivating :\

Kaj Hansen

What do you mean @Committingtoachallenge?

Mike Miller

06:30

@Kaj Did you see the answer to the question?

Kaj Hansen

I see it now that I've navigated back, but I'll hold off on reading it for now so that I still have the challenge of solving it on my own.

Hey there @Alizter

Will Hunting

Hi @Alizter @KajHansen lol.

Kaj Hansen

Hey there @WillHunting

I saw your purple square and started typing @Jas... wondering why it wouldn't autocomplete your name.

Wait....you are JasperLoy, right? I don't want to embarrass myself here.

Will Hunting

Yes, I am the great banana.

Kaj Hansen

:D

Will Hunting

06:41

Yesterday, I found some peace by telling myself there is no need to make sense of some absurd things others do, because they are absurd.

Kaj Hansen

Oh God: math.stackexchange.com/questions/1044584/…

Will Hunting

What about it?

Kaj Hansen

Just the formatting/typsetting is hard to parse.

Will Hunting

I recently learnt the difference between an audio cd and an mp3 cd.

Kaj Hansen

...There's a difference?

Will Hunting

06:47

mp3 files are more compressed, so many more can fit onto a single disc. The sound quality is technically inferior but indistinguishable to the human ear. These also cannot be played by older CD players.

Kaj Hansen

Oh yes, now I remember why I rip CD's with the "Lossless" option enabled.

Will Hunting

I will be using Assimil French With Ease and Assimil German With Ease to study French and German. They come with either audio CDs or mp3 CDs, so I went to look up the difference.

Kaj Hansen

Compression in general can do some pretty weird stuff. In image compression for example, jpg's will often come out "grainy".

In that case it's not that big a deal.

I'm studying French right now :) Are you starting fresh @WillHunting?

Will Hunting

I wonder how many books I will be able to study next year. I have 12 math books, 1 for each month might be a bit too much.

@KajHansen Yes, total beginner. But I won't study them until next next year, lol.

Kaj Hansen

I'm starting my third semester of French in the spring.

Will Hunting

06:55

Some people claim that they learnt more from Assimil than years of school study.

Each book has about 100 lessons. You need about 1 hour per lesson per day.

Kaj Hansen

My problem now is definitely practice.

Will Hunting

After 100 lessons, you are supposedly at B2 of the CEFR.

Kaj Hansen

I feel like I have much of the grammatical structures down and most, if not all of the verb tenses. When I read French now, most of the time it's "I don't know what [vocabulary word] means."

Will Hunting

I really hope I can start studying math successfully next year. I can't afford to wait any longer. Also, I really hope I can get into grad school, otherwise life ain't worth living.

@KajHansen Just get a good dictionary.

Kaj Hansen

:/

Will Hunting

07:00

I am now drinking lots of coffee.

The Assimil books are very expensive. But they seem to be the best way to learn a language, after doing research over 9000 products.

Kaj Hansen

Over 9000! :O

Will Hunting

I won't buy them yet until I have finished my 12 math books, lol.

Kaj Hansen

What math are you working through now?

Will Hunting

I am not studying anything now. I will start on 01 Jan. It is an obsession I have that I must start on that day.

Balarka Sen

Hello @Kaj

So much for the 6 smacks. SHEESH.

Kaj Hansen

07:16

hahaha

Balarka Sen

@KajHansen Have you ever seen a proof of the fact that $\Bbb Z$ can't appear as Galois group of any algebraic extension over $\Bbb Q$?

I have been wondering that for a while. Probably I have read it somewhere, not sure.

Kaj Hansen

I haven't :/

Balarka Sen

well think about it, @Kaj

Balarka Sen

07:41

oh it's not as complicated as i thought. but i'd like a classical way to do this though.

Committing to a challenge

@KajHansen Sorry, I had a nap. What I meant was that I am currently working on Zorich - Mathematical Analysis I, Cohn - Classic Algebra & Axler - Linear Algebra done right. All of these I am on an exercise section, P22(~50 exercises), P29(9 exercises) & P19(15 exercises) respectively

Kaj Hansen

You don't need to do every exercise @Committingtoachallenge. Or is that your challenge?

Also, isn't that Axler book the one that doesn't touch determinants at all? haha

Committing to a challenge

@KajHansen Yep, no characteristic equation, I am excited XD

@KajHansen If I don't do every exercise, I can practically skip learning things xD. Therefore I must do every exercise

Kaj Hansen

Determinants are soo useful. I can't imagine doing lin alg without them.

Is my matrix invertible? Oh? Determinant is zero? Guess not!

Balarka Sen

@Kaj do you have a good galois/field theory problem? i am bored off with these covering space stuff :(

Committing to a challenge

07:46

@KajHansen They are useful, but very boring. I will definitely need them for any optimization or phase analysis

But my linear algebra actual knowledge is atrocious

I am like an engineer in that regard :(I started in Electrical)

Kaj Hansen

I'll try and think of something @BalarkaSen. Nothing right off hand.

Balarka Sen

not so @Committingtoachallenge. Imagine determinants as areas of parallelpipeds with dimensions being the rows of the matrix.

Committing to a challenge

@BalarkaSen I am not sure what you are 'not so'ing

Balarka Sen

"determinants are very boring"

Committing to a challenge

@BalarkaSen When I am checking for non-degeneracy on numerous equations with constraints for my bordered hessian, and I have to do the determinant for $5\times 5$ or greater :\

It is the characteristic equation that is boring I should clarify

Balarka Sen

07:49

tedious, sure. but not boring.

Will Hunting

08:00

@KajHansen You should be sleeping, lol.

Kaj Hansen

You are quite right @WillHunting

Committing to a challenge

@KajHansen What is your timezone Kaj?

Kaj Hansen

EST

It's 3 AM here

Committing to a challenge

@kaj You have finished your finals?

Kaj Hansen

Nope. They start the week after this week.

Committing to a challenge

08:09

@KajHansen How are you feeling for them?

Kaj Hansen

Shouldn't be too bad. I'll probably be studying a lot next week though. Ugh.

Committing to a challenge

@KajHansen For my exams I am usually studying all day every day for the last $2$ weeks, I didn't even log into chat from the second on Nov 2nd to the 19th

Kaj Hansen

I figure I'll start studying on Monday, which gives me around 10 days of studying. The only exam I'm concerned about is real analysis. I'll put most of my time into that.

Committing to a challenge

@KajHansen My most worrying was optimisation theory, I really want my results(which come out in three days)

Kaj Hansen

Good luck!

I can't remember; are you in grad school @Committingtoachallenge?

Committing to a challenge

08:24

@KajHansen No, going into third year next year. But I was originally in Electrical Engineering

@KajHansen The Math I learnt while doing Electrical was bad Engineering Math, which really hurt my knowledge for when I switched over

Kaj Hansen

Ah

LOL. Yeah, I feel you

Robert Cardona

Hi. Trying to work through Hatcher's proof of Borsuk Ulam $n = 2$. Can anyone tell me why $h(s + 1/2) = -h(s)$ implies $\tilde h(s + 1/2) = \tilde h(s) + q/2$ where $q$ is odd?

Committing to a challenge

@KajHansen Specifically Calculus more than anything. "Here is stokes theorem, it is used for flux and curl and stuff", "Memorise the forumla so you can do it quickly"

Kaj Hansen

haha, yeah. I got a lot of that in high school.

Committing to a challenge

@KajHansen My High school was awful

Kaj Hansen

08:27

:/

Committing to a challenge

@KajHansen Our Math was really weak. We learnt some Integration in the final year :\

Kaj Hansen

I definitely had to fill some calculus holes going into college.

Committing to a challenge

@KajHansen Well hopefully over these three months I have off, doing these textbooks will fix me

DanZimm

$test$

Kaj Hansen

$\text{test}$

Committing to a challenge

08:29

$\mathfrak{test}$

DanZimm

$\operatorname{test}$

Kaj Hansen

$\mathcal{Q}$

Committing to a challenge

$\bf \text{ test } $

$$\large \bf \mathbb{TEST}$$

Kaj Hansen

$\text{Do \$ signs work?}$

So what exactly do you do when you have a legit need for the $ sign?

Committing to a challenge

$\text{ signs do work ? }$

$\text { \$ }$

Slash it

$ \$ $

Kaj Hansen

08:31

There we go

Committing to a challenge

Or you can ` it

Kaj Hansen

$ `$ $

Committing to a challenge

$ \text{ so nothing renders } $

Kaj Hansen

Ah

Committing to a challenge

$\frac{12}$

Kaj Hansen

08:33

Almost everything I know about LaTeX I know by trial-and-error on MSE.

Committing to a challenge

$\left\{\begin{align} \LaTeX \text{ is fun !} \\ \text{ :) } \end{align}\right.$

@KajHansen Me too :)

Kaj Hansen

And now I can typeset documents with titles and whatnot nearly as fast as I can write it up.

Committing to a challenge

Yep, I love it. For documents I am using MikTex at home

and meta.math.stackexchange.com/questions/4666/… at uni

Kaj Hansen

I use TeXWorks myself, but personal preference.

Committing to a challenge

with the stack printer

Kaj Hansen

08:35

That's an AWESOME meta post.

What's the stack printer?

Committing to a challenge

It prints an entire stack exchange page, question and answers

Kaj Hansen

Ah

Committing to a challenge

See here stackprinter.com/…

So I just use that on the sandbox if I am at uni, and good to go

Kaj Hansen

That's pretty cool. I had no idea that existed.

DanZimm

is ignorant to any other tex distro besides texworks

Committing to a challenge

08:37

@KajHansen I think I use it more than anyone else(excluding banned Makoto)

Kaj Hansen

Same here @DanZimm

Granted, I've only been a member here for 8 months.

DanZimm

curious, what do y'all study? Haven't ever communicated with this time of the day crowd I dont think

Kaj Hansen

(Is it bad I've visited every consecutive day since I joined MSE?)

DanZimm

I had like a 300 day streak

Kaj Hansen

I'm a lowly undergrad @DanZimm. I like abstract algebra.

DanZimm

08:39

Ah very nice - any specific areas of it?

Committing to a challenge

I have been a member for about 18 months, but I left my primary account as I had asked embarrassingly simple question, I don't care about that now, so I ask whatever

DanZimm

or just generally speaking structure on sets?

Committing to a challenge

@DanZimm I am doing pure math, but I am only 2nd year through

@DanZimm I enjoy Optimisation mainly at the moment

DanZimm

So like calculus of variations type stuff?

Committing to a challenge

@DanZimm Yep

DanZimm

08:40

optimal control theory?

very cool, and that ladder one can be /very/ difficult

Kaj Hansen

I fell in love with Galois theory myself. If I become a professional mathematician one day, I think the inverse Galois problem would be fascinating to crack away at.

Committing to a challenge

Yeah I have done a little bit of optimal control

DanZimm

unfortunately I don't have the background in measure theory to do optimal control theory

Committing to a challenge

The P O Q curves stuff etc

DanZimm

@KajHansen indeed it would ;D

Committing to a challenge

08:41

Or some places call them $C^+ C^-$ curves

DanZimm

@Committingtoachallenge already lost me ;P I just very general qualitative basics of that

And I only know about it because of my interest in calculus of variations

Committing to a challenge

Oh okay haha, well they are the solution curves to the origin when there are nodes

DanZimm

ah right

yes I remember a talk on that

Kaj Hansen

I really like learning as much new stuff as possible. I try to take 2-3 math courses each semester just to see what's out there. I really enjoyed the topology section in real analysis this semester, so I'm looking forward to next semester's point-set.

We've also been doing some super-cool stuff in my ODEs course with Index theory and stability theory. I'd be interested looking into that some more at some point.

DanZimm

(I remember going to the talk not the content in the talk)

Kaj Hansen

08:42

Ultimately, I don't really know what I want to do. I just like learning new stuff.

Committing to a challenge

@KajHansen I have done zero topology, inside and outside of uni. Next semester will change that

DanZimm

index theory O.o - I personally study PDE and haven't ever ran into that

stability I know about though

topology is fun... but unfortunately my course on it skipped my favorite part: sequences/nets

the professor wanted to get to algebraic topology, so I guess I know a little of that

Committing to a challenge

Have you done Rudin or Zorich's textbooks @DanZimm

Or Axler or Cohn?

DanZimm

I've done baby Rudin's analysis book and some of Rudin's "Real and Complex Analysis", heard of Zorich but never done

Committing to a challenge

@DanZimm Was baby Rudin good?

DanZimm

08:46

I enjoyed it thoroughly - it's what really taught me the basics of modern mathematics

Kaj Hansen

@DanZimm, the index theory we're doing is this sort of thing, but not through the lens of differential topology: en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf_theorem

DanZimm

also it's what pulled me from highschool math to college math

Committing to a challenge

Were the first few chapters bad? I have heard that others think they are

DanZimm

(well in the US that is)

@KajHansen huh that's pretty cool - connected analysis of the de to geometry of the space

beginner

hey @kaj !!

Kaj Hansen

08:49

Hey there @beginner

beginner

you are up late again!

which is good for me hehe. could i ask your help for one small question?

DanZimm

@Committingtoachallenge yea the first chapters are what really pulled me - lots of pounding my head during that time, but ultimately for the better I would say. The first chapter isn't bad, the second made me think math wasn't right for me (in the end I guess it was), the third got much easier

@Committingtoachallenge (probably because I had a better mindset by that point) and then I went on from there. I skipped a few chapters in order to get to Measure Theory in the end, but I'm sure I could teach myself what I skipped if I needed to (meaning it wouldn't be the most difficult thing in the world)

Committing to a challenge

@DanZimm How long did the first brutal chapter take you(just roughly)

Kaj Hansen

We're mostly just looking at primarily vector fields in $\mathbb{R}^2$. Essentially what we'll do is draw a closed curve somewhere on the plane and look how the vector field "changes" along that curve. Formally, we define the index of that curve to be $\displaystyle \frac{1}{2\pi} \int_C dw$, where $dw$ is the $1$-form $\displaystyle \frac{fdg - gdf}{f^2 + g^2}$.

DanZimm

hrm probably 1.5-2 months

Committing to a challenge

08:51

@DanZimm But that was in highschool did you say?

DanZimm

keep in mind this is with 0 understanding of proof based math

my first real math class (in college)

Kaj Hansen

I should add that $f$ and $g$ are the $x$ and $y$ components of the vector field. $F = \langle f(x, y); g(x, y) \rangle$.

DanZimm

I knew calculus beforehand and ODEs

and I was a physics major on top of it so I was very used to hand wavy type of arguments, the big thing was what it meant to be rigorous and precise

Committing to a challenge

@DanZimm That sounds alright, I will be doing it after I finish Zorichs Mathematical Analysis I, Cohn - Classic Algebra and Axler - Linear algebra done right

DanZimm

heh, good luck!

Committing to a challenge

08:53

@DanZimm Yeah I am excited for that one. Everyone loves it(bar the first few chapters)

Kaj Hansen

At any rate, the 'index' of the curve ultimately tells you a LOT about the vector field and distribution of fixed points.

DanZimm

I haven't heard of the latter 2 (mostly because I'm not terribly interested in algebra)

Kaj Hansen

@beginner, I might be able to help, lol

Committing to a challenge

@DanZimm Nor was I, I think that is why I need to do them haha

DanZimm

@KajHansen ahh very cool!

unfortunately my ODE teacher was an algebraist and could give a shit about teaching us ODEs correctly

beginner

08:54

i am trying to find an example of a none empty subset of $\mathbb{R}^2$ such that i have closure under addition, and my additive inverse, but i don't have multiplication closure

DanZimm

@Committingtoachallenge heh - linear algebra is quite helpful if you wanna learn functional analysis

Committing to a challenge

@DanZimm Where is your math standing, I don't think I asked

@DanZimm Yeah I am doing functional next semester for Uni, so I really need to get things together

Kaj Hansen

It depends on how you define multiplication @beginner. $(a, b) \cdot (c, d) = (ac, bd)$?

DanZimm

I'm currently a grad student in a masters program - interested in PDEs

Kaj Hansen

And likewise for addition?

beginner

08:55

yes i think so, the dot product way i think

DanZimm

also: classic algebra sounds hard, good luck with that ;P

Committing to a challenge

@DanZimm Awesome, how are you finding it?

@DanZimm It is pretty hard :S

DanZimm

finding the program or PDEs?

Kaj Hansen

Oh careful beginner. Dot product outputs real numbers.

Committing to a challenge

@DanZimm The program

DanZimm

08:56

program = mediocre but I have a couple of great mentors

Kaj Hansen

...Ted?! Awake at this hour?

DanZimm

if you're familiar with PDEs (specifically viscosity solutions) you might of heard of the name Bob Jensen

beginner

oh i made a mistake. i am doing scalar multiplication closure, i will try again

Ted Shifrin

Yeah, @Kaj ... Ate too big a dinner in ATL eight hours ago ... Aand you?

DanZimm

eh Robert Jensen, nonetheless he's one of my mentors

very smart, very funny

Kaj Hansen

08:58

I'm going to sleep soon. Plus I'll have plenty of time to sleep on the long, long drive back into Athens tomorrow.

DanZimm

Question to all: would it be weird to make a wikipedia page for one of your professors?

Kaj Hansen

I've been getting 'nerdsniped' in various ways.

beginner

wow ted is a professor?

Ted Shifrin

Oh?

Kaj Hansen

More importantly, @DanZimm, how do the Wiki denizens decide when someone is important enough to have their own Wiki page.

Transcript for

$Mathematics$ Mathematics