Mathematics

12:45 AM

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

1:07 AM

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

1:22 AM

@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

1:32 AM

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

1:53 AM

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

2:09 AM

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

Karl Kronenfeld

2:30 AM

@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

2:50 AM

@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

2:56 AM

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

3:01 AM

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

3:07 AM

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

can S be infinite?

yes

dang

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

3:10 AM

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

3:13 AM

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

3:18 AM

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

3:26 AM

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