Mathematics

does somebody have any ideas regarding ways to learn mathematics that don't require visuals? Regarding this question: math.stackexchange.com/questions/348697/…

caveman

lol are idding me

learn math on the road

anon

tangent - dubuque connected the idea of schreier refinement in group theory to gcd/lcms in number theory rather strikinlgy here

caveman

2:54 AM

please tell me this is ajoke

Jorge Fernández

no

Why do you think it is not feasible?

caveman

@anon, that confused me because - while J.H does imply FTA - a composition series isn't a factorization

Mariano Suárez-Alvarez

there are blind mathematicians

Jorge Fernández

I don't have any distractions and I have 1 hour

Mariano Suárez-Alvarez

thinking it is unfeasable is silly

caveman

2:55 AM

@MarianoSuárez-Alvarez, do they drive cars

Mariano Suárez-Alvarez

sorry, I am ignoring you

caveman

lol

upset you can't troll me

κρανίοπεριπολία

ouch

Jorge Fernández

@caveman I do not drive the car. I go on bus.

Mariano Suárez-Alvarez

yes

anon

2:55 AM

@caveman it is more than a factorization - it is a factorization where the factors (subquotients) are ordered and have much more structure. we can think of it as doing composition series for Z/nZ in various ways

Mariano Suárez-Alvarez

as everybody knows, I am an eager troll

caveman

I thought it was a filteration - something less than a factorization

Jorge Fernández

If I were driving at the same time I would not do that.

caveman

@JorgeFernández, read a book

Jorge Fernández

I get dizzy

anon

2:57 AM

the quotients of successive terms in a filtration are unique up to order and, conceptually, are thus the "factors" of the group, in a much more general way than integers factoring into coprime things

caveman

ok

Jorge Fernández

No ideas?

not even you caveman? @caveman

anon

I would just use a pen and paper on the bus and do math like at any other time I'm doing math. Actually - not only would I do this, but I occasionally do do this.

Mariano Suárez-Alvarez

@anon, in fact, the uniqueness of simple subquotients of $Z/n Z$ is precisely the uniqueness part of the FTA.

Jorge Fernández

clearly you have not met my bus driver, doing that would imply getting your pen in your eyeball.

amWhy

3:01 AM

@MarianoSuárez-Alvarez Hmmm, good warning. I don't think you've ever trolled me. BTW, I thought I knew what a "troll" was, but as I understood "troll" and "trolling", it seemed to be more of an extreme behavior than what I've been seeing here in chat: reference to "I'm just trolling you" or "trolling me again, heh?" I guess I've never taken you, Mariano, to be trolling, at least as far as I understood the term. So someone, enlighten me: is it "teasing", "trying to get a "rise out of" someone?

caveman

just listen to nice music and relax

Jorge Fernández

@caveman I might as well talk to girls on the bus instead of that.

anon

@amWhy the word has clearly become diluted and varied over time. it used to simply mean deliberately getting a rise out of someone (I think back to usenet, but my familiarity began with 4chan ofc), but now it also applies to people deliberately teasing in a lighthearted manner, or unintentionally being foolish to the point of not being worth others time, or malicious bullying, etc. It is now a spectrum.

Also, I believe Mariano was being sarcastic

Jorge Fernández

I find graph theory, combinatorics, and group theory does not require a lot of visualization

of course the audio needs to go pretty slow.

amWhy

@anon Okay, that fits with my perceptions: that the term has become diluted and now refers to a wide spectrum of behavior. Yes, I do know @Mariano is familiar with sarcasm ;-)

anon

3:08 AM

I've read of blind mathematicians listening to math lectures recorded specifically for them; some of them play the audio at like 10x normal speed, incredibly (here - JDH]

Jorge Fernández

holy crap, I need those records.

Does anyone know a blind mathematician?

Mariano Suárez-Alvarez

I do

I have listened to conferences while in the gym

for example, there is a series of conferences at the College de France, by people like Alain Connes.

anon

related - ams.org/notices/200210/comm-morin.pdf

Mariano Suárez-Alvarez

they write stuff on the blackboard, but not being able to see it does not hurt much

Jorge Fernández

fascinating www.ams.org/notices/200210/comm-morin.pdf

nvm

Mariano Suárez-Alvarez

3:11 AM

@anon :-)

anon

@JorgeFernández ah, the magic of identical google searches

Jorge Fernández

although it looks fancier when you make it into a hyperlink.

anon

@MarianoSuárez-Alvarez are you guys related or something?

Jorge Fernández

Álvarez is pretty common and one of them lives in Argentina and the other one in Spain.

anon

interesting

Mariano Suárez-Alvarez

3:18 AM

indeed

robjohn

@anon no sweat

amWhy

3:39 AM

@anon Thanks, btw...I made two stupid errors tonight... :-/

user1

Wrong tab. :)

amWhy

@robjohn I was on the look out for "bunny ears" on an orange meanie on Easter Day ;-)

You disappointed me :-(

robjohn

@amWhy I was not around on Easter. We were out almost all day.

amWhy

@robjohn I didn't think I saw too much of you...But you have to admit, it would have been a nice "touch" to complement the "santa cap" at Christmas time ;-)

κρανίοπεριπολία

hi

caveman

3:51 AM

hi

let us have two composition series for a group G with penultimate terms K and L

we could show they are equivalent by induction

so G/K and G/L are simple

since $K \lhd KL \lhd G$ this forces $KL = K$ or $G$

I don't understand that

why is that

user1

KL/K is a normal subgroup of G/K.

caveman

thanks user1

similarly KL = L or G

so if KL is not G then KL=K=L

we delete it and are done by induction

if KL is G, then

$$\frac{K}{K \cap L} \simeq \frac{G}{L} \simeq \frac{G}{K}$$ by the second isomorphism theorem mathworld.wolfram.com/SecondGroupIsomorphismTheorem.html

κρανίοπεριπολία

@Ethan Hi (removed)

user1

The last isomorphism is false.

caveman

aww thanks @user1, why is it false?

user1

3:58 AM

For instance let $G=\mathbb Z_2\times\mathbb Z_3$ and take composition series.

@caveman In general, $G/K\cong KL/K\cong L/(K\cap L)$.

Rather than $K/(K\cap L)$.

caveman

I see

well I'll have to try to correct my proof here

caveman

4:12 AM

my notes say take a composition series for $K \cap L$

so you have

$$1 \lhd \cdots \lhd K \cap L \lhd L \lhd G$$ and $$1 \lhd \cdots \lhd K \cap L \lhd K \lhd G$$

user1

I'm sorry, what is your goal?

caveman

prove that all composition series are equivalent in the sense that their factors are a permutation of each others

the Jordan Holder theorem

user1

Ah, ok.

This will do it then.

caveman

how?

user1

Are you inducting on length of composition series?

caveman

4:14 AM

yes

user1

Just account for composition factors. Compare the given composition series for $G$ with penultimate term $L$ to the first listed here. Compare the two listed here.

caveman

they have the same factors until the last two, wihch are $L/(K \cap L)$, $L/(K \cap L)$ or $K/(K \cap L)$, $K/(K \cap L)$

user1

After substituting isomorphic groups, yes.

caveman

starting with $$1 \lhd \cdots \lhd L \lhd G$$ and $$1 \lhd \cdots \lhd L \lhd G$$ I can intersect with to get $$1 \lhd \cdots \lhd L\cap K \lhd K$$ and $$1 \lhd \cdots \lhd L\cap K \lhd L$$ but those aren't composition series.. and I can take a composition series for $L \cap K$ to get composition series $$1 \lhd \cdots \lhd K \cap L \lhd L \lhd G$$ and $$1 \lhd \cdots \lhd K \cap L \lhd K \lhd G$$ which are equivalent up til the last two

but I don't see how can I say my starting two are equivalent

user1

A bit too complicated of an approach, I think.

caveman

4:22 AM

how should I prove it instead?

user1

@caveman These two are equivalent, by the second isomorphism theorem.

caveman

oh!

because L and K are normal

LK = KL

user1

Now, we want to show that $1\lhd \dots\lhd L_0\lhd L\lhd G$ is equivalent to $1\lhd\dots\lhd L\cap K\lhd L\lhd G$.

caveman

yeah, that's important - I need to show that

oh it's immediate by induction

user1

Yes. You chop off the last part, and have series for $L$, which are necessarily equivalent.

caveman

4:25 AM

ok thanksa lot I have a complete proof now!

and it doesn't use the Zassenhaus lemma!

Fun stuff. :)

thanks for your help

@caveman No problem.

4:40 AM

this proof is very strange I'm confused about it

it seems to prove the stronger result that the factors are isomorphic in order

not just a permutation

user1

How so?

caveman

because we showed that $G/K \simeq G/L$

user1

When?

caveman

we have $G/K\cong KL/K\cong L/(K\cap L)$

oh wait

I've misunderstood something

I see it now!!

the last two are swapped around

user1

Indeed. This shows in a convoluted way that every permutation is a product of transpositions.

amWhy

4:53 AM

@κρανίοπεριπολία hello!

κρανίοπεριπολία

@amWhy hi

amWhy

@κρανίοπεριπολία How are you?

κρανίοπεριπολία

@amWhy Fine thanks. How are you?

amWhy

@κρανίοπεριπολία Okay...Will go to bed soon. Just turning up "empty" on math.se...

κρανίοπεριπολία

...when you awaken there should be plenty of new quiries.

amWhy

4:58 AM

@κρανίοπεριπολία Yes...there were a few tonight...and I answered a handful, but not much in the way of "upvotes"...oh well, a new day ahead! ;-)

user19161

@caveman Mariano was going to help you, but then you were sarcastic to him it seems. There are several Jordan Holder theorems.

user19161

It really does not pay to be kind. One gets scolded for trying to help people.

user19161

This is really so amazing.

Jayesh Badwaik

@JasperLoy Chill. Caveman's loss.

2

amWhy

5:14 AM

Hi, @Jayesh !

Jayesh Badwaik

@amWhy Hello. :-)

Wazza? (wassup?)

amWhy

Huzzah? Wuzzup? Not too much. But hangin' in there, and ready for change (+)...

(+): in a good way ;-)

@JayeshBadwaik How about you?

Jayesh Badwaik

Good. :-)

Samuel Handwich

nobody makes good cigarettes north of the mason dixon line.

amWhy

@JayeshBadwaik You're always on the bright side! I like that... :-)

Jayesh Badwaik

5:17 AM

@amWhy Well, currently, I am enjoying the happy life as I have it right now.

@amWhy Thanks. :-)

@SamuelHandwich cigarrettes are never good. :-)

caveman

@SamuelHandwich, why's that

Samuel Handwich

@JayeshBadwaik foolishness.

@caveman maybe the climate.

maybe a weak, sun-deprived spirit.

Jayesh Badwaik

@SamuelHandwich Why? Ham sandwiches are much better anyway

Samuel Handwich

@JayeshBadwaik which cigarettes have you smoked?

amWhy

funny, @Jayesh!

Yes, I want to say Hamuel Sandwich when I see the name!

Jayesh Badwaik

5:24 AM

@SamuelHandwich I have never smoked. I just wanted to say that cigarette smoking is not good in general.

Samuel Handwich

@JayeshBadwaik i figured as much. you should try it. it's good for you.

caveman

lol

Jayesh Badwaik

@SamuelHandwich Ahh, no thanks. :-)

cosmicraga

6:07 AM

I found it hard to find many online study materials and particularly codes which are simple enough (not full large packages) as for the purpose of demonstration of the underlying concepts of Monte Carlo simulation techniques in the field of statistical mechanics.

It would be nice to have your help in this regard to find those materials online.

Thanks.

Vrouvrou

7:11 AM

hi

is there anyone ?

κρανίοπεριπολία

8:47 AM

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Transcript for

$Mathematics$ Mathematics