Mathematics - 2015-07-19 (page 2 of 3)

Remember me

06:18

Finished chemistry at last ....

skill patrol

Organic?

Remember me

Quantum chemistry @skill

skill patrol

I see.

Soham Chowdhury

Quantum chemistry: "okay, so here are Bohr's postulates; here's why they suck; here are better ones . . . " and then, suddenly, out of nowhere, you suddenly get Schroedinger's equation. >_<

Remember me

And the worst part ..... how are orbitals they way they are @Soham

Soham Chowdhury

06:32

I'm having trouble uploading my pic for KVPY.

Remember me

Someone told me look at the solutions and roots of the Schrodinger equation...

Balarka Sen

@SohamChowdhury sure, that's part of classification of p^3-order groups

Remember me

I have no idea how you get the solutions

Soham Chowdhury

It says it's too big. It's f**king 600 KB. this is 2015, India.

Remember me

@Soham Zip it up

Compress and zip it .. will work I guess

Soham Chowdhury

06:33

@BalarkaSen I went back to the first chapter of Aluffi. it apparently has pushouts and pullbacks. an exercise in ch. 2 asks to find those in $\sf Ab$. interesting.

@Rememberme .jpg only, no zips allowed

do you know what those are in $\sf Set$?

Remember me

Well then call them up

Balarka Sen

@SohamChowdhury Schroedinger equation is motivated from the classication differential equation for strings (classical strings, not those fancy stuff).

Soham Chowdhury

I know. I used to be mad about physics once.

My mother has a whole book on non-fancy strings.

Remember me

@Balarka you know the solutions for the equation?

Balarka Sen

nope. I just know it exists

Remember me

06:35

And what is 4 fold @Balarka

Balarka Sen

huh?

Soham Chowdhury

there's different equations for plucked strings and for vibrations of bowed strings like on violins. :)

Remember me

Yes it says that the f orbital is 4 fold or complex .. what do they mean by 4 fold..

Soham Chowdhury

so, so cool. but too many assumptions.

Balarka Sen

dunno

Soham Chowdhury

06:37

did you see my ping about pushouts?

Balarka Sen

yes

Soham Chowdhury

and?

Balarka Sen

I am not going to tell you what they are.

Remember me

Yes that is why sometimes I end up hating chemical bonding... Tell me is there any proof that hybridization actually happens @Soham

Balarka Sen

No, but there are evidence that it happens.

Soham Chowdhury

06:38

is* evidence

all of science is based on repeated experimental verification and a lack of credible alternative explanations.

so, you know.

Remember me

Yay!!!

Soham Chowdhury

@BalarkaSen I mean, you've seen that before?

in $\sf Set$ as well?

Balarka Sen

sure

Soham Chowdhury

what was I expecting

Remember me

My form was successfully submitted

Soham Chowdhury

06:40

how big is your pic?

:(

Balarka Sen

@SohamChowdhury I don't know what you were expecting.

Remember me

wait let me see

13.37 KB@Soham

Soham Chowdhury

@BalarkaSen no, I mean it was stupid to ask you if you've seen it before. :P

@Rememberme how the hell?

Balarka Sen

pushouts and pullbacks are standard objects. I learnt about them while I was fiddling with solenoids.

Remember me

@Balarka $S^1\times S^1$ is a disk right and gluing the ends give me $S^2$ which is a sphere

Balarka Sen

06:42

$S^1 \times S^1$ is not a disk.

Remember me

It is not .... then what is it

Soham Chowdhury

torus, obviously

Balarka Sen

Stick a circle at each point of another circle.

You get a torus

@Soham Obviously? Now I would ask you to prove that!

Soham Chowdhury

but considering you had trouble with $\Bbb R \times S^1$ yesterday . . . :P

@BalarkaSen you type too fast for me to add a disclaimer.

Balarka Sen

That is, prove that $S^1 \times S^1$ is homeomorphic to the subspace of $\Bbb R^3$ we call "the torus".

Remember me

06:43

Yes I am having trouble with these stuffs ... SO that is why I am considering as many examples as in can

Soham Chowdhury

@BalarkaSen well, I think I know a continuous bijection either way. you can parametrize all points on the torus by two angles.

Balarka Sen

homeomorphism $\neq$ continuous bijection :P

Soham Chowdhury

with cts inverse

"either way"

Balarka Sen

Well, you think you do. Now parametrize the torus and write it down :P

I'm going to do that to you everytime you say "obvious"

Soham Chowdhury

:(

Bulerka pls

Remember me

06:45

@Balarka $(\Bbb{R}\times S^1)\times (\Bbb{R}\times S^1)$ what will this give me ?
Two infinitely long cylinders attached with each other...

Soham Chowdhury

what crap

Balarka Sen

Something not easy to visualize.

Soham Chowdhury

@BalarkaSen wait.

@Rememberme a cylinder sticking out of each point of a cylinder? :P

Remember me

okay I will be right here
$(0,1)\times (0,1)$ is a circle right

Balarka Sen

No, it's not.

Duh.

Soham Chowdhury

06:46

you need identifications.

that's an open square

Balarka Sen

It's an open square.

Remember me

ahh now i get it

Soham Chowdhury

Munkres ch. 1?

Remember me

Glue the ends you get a cylinder glue the ends of a cylinder you get a torus

And that is just two circles sticked to each other

Balarka Sen

I don't think you can glue ends of something open.

Soham Chowdhury

06:48

how do you do the torus thing without a little explanation?

Balarka Sen

What Soham was saying about identifications was silly.

He doesn't know quotient spaces, don't pay attention to him.

2

Soham Chowdhury

there aren't any ends :P

sorry

Remember me

oh :p

Soham Chowdhury

I starred, btw

as a monument to my ignorance

Remember me

@Balarka So how do you get a disk ...

Balarka Sen

06:49

An open disk or a closed disk?

Either way, $(0, 1) \times (0, 1)$ and $[0, 1] \times [0, 1]$.

Remember me

A closed disk... closed means the usual definition of closed right?

Balarka Sen

Prove that $[0, 1] \times [0, 1]$ is homeomorphic to the closed disk, btw.

And by "prove" I mean "write down a homeomorphism".

@Soham So, what kind of algebra are you doing?

dREaM

@BalarkaSen can we use polar bears?

Soham Chowdhury

@BalarkaSen revising the entire groups chapter, right from subgroups and everything.

Balarka Sen

cool. have you done much linear algebra, though?

Soham Chowdhury

06:52

e.g. "show that $C_2 \times C_3 \ncong C_2 * C_3$". this one was so cool.

Balarka Sen

yep.

Soham Chowdhury

@BalarkaSen I know a little. say Artin ch. 3

@BalarkaSen you know how the exercise does it?

Balarka Sen

I forget what ch. 3 is about.

Soham Chowdhury

"Vector Spaces"

Remember me

Oh so you haven't done lin.transformations and functionals yet

Soham Chowdhury

06:53

show that there are injections $C_2, C_3 \hookrightarrow S_3$

Balarka Sen

oh, I see. can you classify all subspaces of R^n explicitly, then?

Remember me

The famous @Balarka question ....

Balarka Sen

@SohamChowdhury Nope. But I think you can do it by noting that $\Bbb Z_2 * \Bbb Z_3 \cong \mathsf{PSL}_2(\Bbb Z)$. I'd have to think.

Soham Chowdhury

maybe. I'm going to do that chapter properly too, then I'll get back to you. right now, no.

Balarka Sen

Meh, C_2 * C_3 is infinite.

I was being silly. No need to go through all that.

Soham Chowdhury

06:56

also, the next one is cool.

Balarka Sen

C_2 x C_3 is finite, while C_2 * C_3 is infinite. So surely there cannot be any isom.

Soham Chowdhury

find a surjection $\Bbb {Z*Z}\to C_2*C_3$.

Balarka Sen

3.6. is just killing a fly with an gun.

Soham Chowdhury

@BalarkaSen ah, but we haven't proven the infiniteness yet

Balarka Sen

@SohamChowdhury this one can be done by noting that the latter is PSL_2(Z)

Soham Chowdhury

06:57

or by looking at words.

Balarka Sen

@SohamChowdhury C_2 * C_3 \cong PSL_2(Z)

Soham Chowdhury

I'm congruent to the modular group?

Balarka Sen

PSL_2(Z) is infinite (why?). done.

Soham Chowdhury

I'll do all the matrix group exercises today.

but the proof of 3.6 is cool, c'mon.

Balarka Sen

it's just silly.

Soham Chowdhury

06:59

well.

Remember me

@Balarka if a set is path connected does it tell you anything about its closure and complement

Soham Chowdhury

i'm far more curious about the pushouts question. (in Ab)

Mike told you that it doesn't (re the latter), @Rem.

Remember me

@Soham how do you get those pdf excerpts

Soham Chowdhury

from a pdf

Remember me

Mike told me about compactness @Soham

Soham Chowdhury

07:00

oh, ok

Remember me

I am asking about a closure

No i mean how from the pdf?@Soham

Balarka Sen

@Soham pullbacks and pushouts both has nice interpretations in AbGrp

Soham Chowdhury

I'm so, so curious. Can't wait to find out. Any hints on how to approach the problem?

Balarka Sen

do you want me to reveal it? (can't think of any hint than just looking at the defn)

Remember me

@Soham what are these AbGrp you keep on talking about

Soham Chowdhury

07:03

@BalarkaSen NO

Balarka Sen

ok.

Soham Chowdhury

@Rememberme category of abelian groups

Balarka Sen

pullbacks are easy to find.

Soham Chowdhury

ok, ok

Remember me

category what are those.. Is my current group theory knowledge enough to understand them

Soham Chowdhury

07:04

unnecessary

Balarka Sen

@Soham do you know what pullbacks are in Set?

Soham Chowdhury

@BalarkaSen I think so. that f(a) = g(b) thing?

Balarka Sen

did you google that?

or is it in Aluffi?

Soham Chowdhury

I googled that, a long time back. when I was doing that first chapter.

@BalarkaSen last exercise of ch. 1

Balarka Sen

oh. yes, that's the thing.

it comes pretty naturally from just the pullback diagram

Soham Chowdhury

07:06

I should verify it, though, just to be sure.

Balarka Sen

try to figure it out by yourself

Soham Chowdhury

Good point, reminding me.

Remember me

@Dream thanks for reminding me the proof which I gave you yesterday...

Soham Chowdhury

@Rememberme you don't need group theory, in principle. I learned what categories were before I even knew the phrase "abstract algebra".

because I was learning Haskell

Remember me

Oh..

Soham Chowdhury

07:07

it's an awesome programming language nearly built on category theory

dREaM

@Rememberme No problem, it's good to go back to the basics.

Balarka Sen

@Remember A category consists of a class of objects, a class of morphisms between objects, such that you can "compose" morphisms and that "composition" has desirable properties like associativity and having an identity.

dREaM

@SohamChowdhury It's awesome when you aren't coding with it.

Soham Chowdhury

I differ.

Remember me

Why do we need categories @Balarka what does it help in?

dREaM

07:09

No, you beg to differ, whether you get the chance to differ or not is entirely up to me

Soham Chowdhury

:P

Balarka Sen

@Rememberme It's the highest level of abstraction you can get. For example, people try to do prove more general things in groups by looking at proofs in arbitrary categories.

Soham Chowdhury

no, n-categories are the highest level. runs :P

Balarka Sen

Categories have a lot less structure, so you can ignore the unnecessary things you need and concentrate on what you want to look for.

But personally, I've never had a feel for categories.

Soham Chowdhury

@Rem: reddit.com/r/math/comments/1mymf6/… is a good example of what Balarka is talking about.

Balarka Sen

07:11

@Soham n-categories are just a generalization of categories.

dREaM

@SohamChowdhury where is that phrase from?

Soham Chowdhury

@dREaM are you begging me? :P

Remember me

Oh okay.. so what i infer from this is that category theory doesn't require any prerequisites ?

Soham Chowdhury

yep

Remember me

It is just basic abstractions of stuff

Balarka Sen

07:12

@SohamChowdhury I don't agree

Soham Chowdhury

but most people recommend knowing a lot of math in advance

it doesn't require anything.

it's up to him, though.

Balarka Sen

If you're going to learn (\infty, 1)-toposes without knowing what groups are, you'll probably have to stop doing math.

@Rememberme No, it is not "basic" abstraction.

Soham Chowdhury

not toposes, but it doesn't hurt to know a few basic defns. anyway, this argument is pointless.

Mike Miller

i don't really see the point of learning how to generalize your examples unless you have examples

Balarka Sen

I agree with Mike.

Mike Miller

07:14

but i agree that this is pointless

Remember me

I am not learning category theory .. Just asking out of curiosity

Mike Miller

but in life, what isn't?

Balarka Sen

hehe.

Remember me

True @Mike

Balarka Sen

@MikeMiller you're having the existentialism-like crisis, or is that message just a joke?

Soham Chowdhury

07:15

Mike Miller

i don't do crises

3

Soham Chowdhury

too mainstream?

Remember me

I feel that I am the most foolish person doing maths here .. who cant even tell that $S^1\times S^1$ is a torus :p

Soham Chowdhury

oh, shut up. I can vouch for the fact that I feel stupider, for larger swaths of time.

:P

Remember me

Hey @Soham You haven't made mistakes that I have made ...

Soham Chowdhury

07:18

I might not have told you.

Remember me

Ask Balarka .. how horror mistakes I have made

Soham Chowdhury

anyway, brb thinking about pullbacks.

Balarka Sen

now you two are fighting for determining who's the stupidest?

how lame

Remember me

hehe

dREaM

@Rememberme I thought it was a doughnut

Remember me

07:21

@Balarka lets say that $\forall x,y\in X$ there exists z such that $x <z<y$ then can I say that no two disjoint open sets can have their union as X

@dream which country are you from?

dREaM

guess

I'll answer yes or no questions

Remember me

Hmm ...

Is it a country whose name starts with A?

dREaM

no

Remember me

E?

dREaM

nah

Remember me

07:24

Okay lets start with continents first

dREaM

ok

Remember me

Is it Europe?

dREaM

no

Remember me

North America

dREaM

yes

Remember me

07:25

Good

Is it a country whose name starts with U ?

dREaM

no

Remember me

Or C?

dREaM

nah

what happened?

Remember me

Thinking

Starts with M?@Dream

dREaM

YES!

Remember me

07:29

Mexico... I guess

dREaM

correct!

Now let me try

Remember me

Fine..

@Balarka chat.stackexchange.com/transcript/message/22870738#22870738 Is this reasoning fine?

dREaM

It took you 7 guesses by the way

Remember me

yup

dREaM

Is your country in Europe?

Remember me

07:31

nope

Chris's sis the artist

@robjohn the integrals I produced this morning are shockingly beautiful.

dREaM

Is your country in Asia?

Chris's sis the artist

I'M COMPLETELY AMAZED!

Remember me

yup

dREaM

Oes your conuntry speak one of the following: English, korean, japanese, chinese?

Remember me

07:32

Yup

dREaM

Oes your country have a population exceeding one billion?

Remember me

Yes

dREaM

Was the character Rajesh Koothrappali from the popular television series: "The Big Bang Theory" born in your country?

Remember me

correct

dREaM

your country is India.

Remember me

07:34

yup

dREaM

yay :)

Remember me

okay gtg

dREaM

all righty, thanks for letting me play

skill patrol

Later pal

Khallil Benyattou

07:59

Can somebody explain the wording of part of this proof?

The part that's getting me is the last sentence.

What does it mean by 'if so are $h_1$ and $h_2$'?

A Googler

08:12

Hello

We normally find area of a curve by slicing parallel to y or x axis

is there any way to find it by slicing parallel to arbitrary line?

Khallil Benyattou

I guess you could perform a change of basis so that you're no longer working with respect to $e_1 = (0,1)$, $e_2 = (1,0)$ @AGoogler.

(Not sure how one would do that though)

A Googler

@KhallilBenyattou Sorry , but i don't know what basis or e means

you mean like rotating coordinate system?

Khallil Benyattou

Yep, that's what I mean.

r9m

@Rememberme hey there ! I'm at Bengaluru right now :)

@Chris'ssistheartist hi :)

just saw the recent monster integral! :O Insane!!

skill patrol

08:34

Hi pal

r9m

@skillpatrol hey!

skill patrol

How are you @r9m?

Soham Chowdhury

@BalarkaSen okay, done the torus thing. it's basic calculus-y parametrization, I realised after a while.

$$\begin{align*}x &= (R + r \cos \varphi) \cos \vartheta\\
y &= (R + r \cos \varphi) \sin \vartheta\\
z &= r\sin\varphi\end{align*}$$
where of course $0\leq \varphi, \vartheta\leq 2\pi$.

Chris's sis the artist

@r9m Salut :-)

r9m

@skillpatrol fine! :-) How about you pal? :)

Soham Chowdhury

08:48

@r9m, are you Bengali?

r9m

@Chris'ssistheartist Hello :)

@SohamChowdhury yesh

Soham Chowdhury

there are four Bengalis on this chat, then. I find it weird. :P

you're from Kolkata?

r9m

yesh

Chris's sis the artist

@r9m Hello:-)

r9m

@Chris'ssistheartist I saw the integral you posted! looks insane!! :O

Soham Chowdhury

08:50

baah. care to share which school you went to? (standard acronyms are fine)

r9m

@SohamChowdhury sphs

Chris's sis the artist

@r9m hehe, just a bit. :-)

Soham Chowdhury

ooh. I went there for nursery. :P

now DBPC.

r9m

okay! :-)

@Chris'ssistheartist a bit?!!! That's crazy!

nyway BBL .. (lunch)

Chris's sis the artist

@r9m :D

Agawa001

09:01

bengali ?

i thought u indian

people mistake my name for indian when they misspell it as "agrawal" (wonder wats that standin for)

Remember me

Bengali are Indians @Agawa001

Agawa001

no @Rememberme bangladesh isnt india

Remember me

I am a Bengali.. Have you ever heard west Bengal...@Agawa001

It is in India

Agawa001

hmm, thats new to me

i thought bangladesh is whole independant country

Remember me

West Bengal was divided into Bangladesh (independent country) and west bengal(in India)

Agawa001

09:19

are you indian rememberme ?

Remember me

Yes

skill patrol

09:33

@r9m fine thanks, enjoy your lunch.

Balarka Sen

@KhallilBenyattou It means $h_1 - h_2 \in H$ if $h_1, h_2 \in H$

Remember me

@Balarka a disk is homeomorphic to a sphere right ?

So if I just prove that [0,1]\times [0,1] is homeomorphic to S^2 then it does the job right?

Agawa001

09:52

@Rememberme what does mean agrawal in hindu ?

Remember me

@Agawa001 it is just a surname

r9m

@Agawa001 wiki

@Rememberme yo there! :) I'm in your city!

Remember me

Nice @r9m

Agawa001

i thought i got enough of these awful limits @Chris'ssistheartist

r9m

:22871537 wow! looks interesting .. :) wait a bit I'll show you something you I bet you'll like unless you knew it already ..

Remember me

09:58

@BalarkaSen sorry for the earlier message but the unit disk is not homeomorphic to S^2

@r9m you in yehalanka ?

Chris's sis the artist

@r9m This morning I developed a new generation of amazing integrals, simply too amazing to be real. Not what I showed you.

r9m

@Rememberme nah (will go there tomorrow) .. near Gandhinagar ryt now ..

Transcript for

$Mathematics$ Mathematics