Mathematics - 2015-05-19 (page 3 of 6)

Destiny freedom

11:00 AM

someone see my problem? maybe is odd problem,But I can't solve it

octatoan

That's what I do.

Remember me

Balarka is it fine if I just write that since according to the question every neighborhood should be $\leq$2 there will be points in the $\epsilon$ ball which will be the same as the points in the closed set@BalarkaSen(sorry if I am annoying you)

Balarka Sen

Proof, proof. you're trying to show that there is at least one element in $B_\epsilon(x)$ which is also in $S = \{z : d(0, z) \leq 2\}$. good idea. but you have to prove that such a thing exists.

hint : use the triangle inequality axiom of the metric $d$. you can do all of this by just drawing.

Remember me

@BalarkaSen sorry I will talk later ... I have a test and will try to do something with you hint

Balarka Sen

why're you thinking about exact sequences, though, @iwriteonbananas? are you doing that splitting-of-homology thingy at the back of chapter 2.2. in Hatcher?

iwriteonbananas

11:16 AM

@BalarkaSen i wanna do mayer vietoris sequence

Balarka Sen

ps : i agree that i was talking nonsense about the exact sequence of Z/2-modules, but in my defense i was sleepy :P R is a free R-module over itself, trivially.

iwriteonbananas

no worries

im blanking right now, can u tell me what i'm missing:

user147690

A mouth I think

Balarka Sen

@iwriteonbananas ok. it's nothing very complicated. just a fancy way to combine excision and long exact sequence

but very powerful indeed.

iwriteonbananas

we're in the situation from before. we have an exact sequence of $\Bbb{Z}$-modules $$0\to M\to N\to \Bbb{Z} \to 0$$. call the second map $f$, the third $g$

define the section $s: \Bbb{Z} \to N$ as you said. map $1$ to something that's mapped onto $1$ by $g$

then linear extension

Balarka Sen

11:18 AM

right

user147690

Direct product of... modules?

iwriteonbananas

i'm trying to explicity write down an isomorphism $\phi: M\oplus Z \to N$. define it by $(m,k) \mapsto f(m) + s(k)$

it's injective since $f$ is injective by exactness, and $s$ is injective by construction (it has left inverse)

why is it surjective?

user147690

It is surjective since N onto Z is the quotient map? E.g. $N/M \cong \Bbb Z\implies N\cong M\bigoplus\Bbb Z$

Balarka Sen

write down an explicit inverse

so that you'll have a general nonsense proof of it. no need to check for injectivity, surjectivity each time.

@AlexClark yes, it's like direct product of vector spaces

well, that's what @iwriteonbananas wants to prove.

user147690

Ahhh the explicit map

user147690

11:23 AM

Universal property thingy

iwriteonbananas

$N\to M\oplus \Bbb{Z}$, $n\mapsto (\text{something},g(n))$

Balarka Sen

?

@iwriteonbananas right, and that something should be determined by your section.

user147690

@BalarkaSen David showed me the thing when he was talking about the universal property 2 months ago

Balarka Sen

fact : having a section N --> Z is equivalent to having a section N --> M

this is one of the things splitting lemma says

I don't know what kind of universal property you're talking about. You mean that the middle group is uniquely determined by the two groups by it's side? @AlexClark

user147690

@BalarkaSen Nah he was teaching me about the universal property in regards to category theory on Vect and when he was covering direct product in regards to category theory

iwriteonbananas

11:26 AM

ahhh

Balarka Sen

oh, that

iwriteonbananas

$\text{something} = f^{-1}(n-s(g(n))$

user147690

@BalarkaSen He showed the explicit map

Balarka Sen

@iwriteonbananas yeah, i guess.

yes, that works

iwriteonbananas

it's well defined since $n-s(g(n))$ is in the kernel of $g$, i.e. in the image of $f$

user147690

11:29 AM

Wow Paul is actually not even logged in here

user147690

Is he... dead....?

user147690

Can you tell me a little about the following list?

Field $\subset$ Euclidean domain $\subset$ PID $\subset$ UDF $\subset$ Integral domain $\subset$ Commutative ring with unit

user147690

So each thing on the left is all of the right but more?

user147690

@BalarkaSen What is a UDF?

octatoan

UFD

Unique factorization domain

user147690

11:35 AM

Ahhh okay

Balarka Sen

@AlexClark I was away. What d'you want to know?

user147690

@BalarkaSen Oh I see the wiki list now

Balarka Sen

k. have you been studying any of the inverse limit stuff we discussed about?

user147690

@BalarkaSen Not yet sorry, I was trying to get rings down and I am pretty deep now: alexpclark.com/index.php/2015/05/18/…

user147690

But I wanted to show that the set of all nilpotent elements of a com ring form an ideal[don't tell me how haha]

Balarka Sen

11:39 AM

oh, sure, rings are good. you need to know about them do anything with inverse limit of rings.

d'you know about quotient rings yet?

user147690

@BalarkaSen Nilpotent elements form an ideal since if we have some $x$ that is nilpotent, we have that $x^m=0$ for some $m\in\Bbb Z^+$, and if we take any $r\in R$ $rx$ is nilpotent since $(rx)^m=r^mx^m=0$ right?

Balarka Sen

yeah.

user147690

So done?

Balarka Sen

yes

user147690

wut

user147690

11:47 AM

Why am I so retarded lol

user147690

@BalarkaSen Not really, should I learn them now?

Balarka Sen

if you want.

user147690

@BalarkaSen Is this to do with the extension of the first isomorphism theorem?

Balarka Sen

you can use quotient rings to generalize isomorphism theorems, sure

user147690

@BalarkaSen Oh wait I have only showed the multiplicative side of things, I need to show that they are closed under addition

Balarka Sen

11:52 AM

oh, you didn't show that $I$ is a subgroup of $(R, +)$?

user147690

@BalarkaSen Not yet

user147690

Will that be hard?

Balarka Sen

nah.

user147690

Okay I'll give it a shot

user147690

12:15 PM

@BalarkaSen Okay I got it, but it is not pretty, perhaps there is a way without binomial theorem?

Balarka Sen

Not that I know of. You just consider $(x + y)^{m+n}$ to prove it.

user147690

And you consider two cases, $m+n$ is even and $m+n$ is odd?

Balarka Sen

yeah.

user147690

Then show the middle part of the sum for even is $0$ and the middle two for odd is $0$

user147690

Ok

user147690

12:17 PM

Then I just need additive inverse

Balarka Sen

Well, you can get round the parity thing by considering $(x + y)^{m + n + 100}$, say.

But whatever.

user147690

@BalarkaSen Yep that is quite clean for sure

user147690

Additive inverse follows as soon as it is an even power great enough

user147690

$x^m=0$ then if $m$ is odd then $(-x)^{m+1}=0$

user147690

if $m$ is even then $(-x)^m=0$

Balarka Sen

12:18 PM

thumbs up

user147690

Sweeet

user147690

I have a feeling this assignment is the easiest we have been given

user147690

That or it gets harder

user147690

@balarka Is showing the gaussian integers are a PID hard, or finding a nec and suf condition on $x\in\Bbb Q$ such that $e^x$ converges in the p-adic norm? They are the last two questions, I might ask you for ideas in a week(but I'll only ask if I am dying haha)

Balarka Sen

no, I don't think it's hard to show that $\Bbb Z[i]$ is a PID. the next exercise can be a pain in the neck, though.

by "pain", I mean tedious, not hard.

octatoan

12:22 PM

@AlexClark what homework is this? Abstract algebra?

Balarka Sen

Have you encountered prime ideals yet?

user147690

@BalarkaSen I know the definition but that is all

user147690

@octatoan Yep

Balarka Sen

@AlexClark That's the only think I really care about in ring theory, to be frank.

user147690

@BalarkaSen Prime ideals or definitions?

Balarka Sen

12:23 PM

Prime ideals :P

user147690

@BalarkaSen Really???

Balarka Sen

yeah, definitely. almost all of algebraic number theory and commutative algebra emerges from studying prime ideals.

user147690

@BalarkaSen well I understand a prime ideal to be an ideal such that having $a,b\in R$ and $ab\in I$ means that either $a$ or $b$ are in $I$(or both)

user147690

@BalarkaSen Wow that is impressive

Chris's sis

@robjohn I highly recommend you 11821 here mat.uniroma2.it/~tauraso/AMM/amm.html. It's from the American Mathematical Monthly.

@r9m btw, did you finalize 11821 (in the spirit of the art :D)? Let me know if you did it. ;)

Balarka Sen

12:28 PM

@AlexClark I guess the alg number theory connection won't be accessible to you yet (I guess you'll encounter all of them later) but I guess something you can reach for at this moment are spectrum of rings. $R$ be a commutative ring. Let $\text{Spec} R$ is the set of all prime ideals in $R$.

Chris's sis

@BalarkaSen see the problem above is strongly related to the number theory. Some insights? Not now, but in some weeks (or whenever you want).

Balarka Sen

Fact : You can topologize $\text{Spec} \, R$.

octatoan

What's a nice abstract algebra book? (I'm familiar with some basics.) Is Aluffi nice? I sort of love the category-theoretic approach he uses from the beginning.

I know Alex here is using DF

And I've done a few chapters of Axler -- do I have to finish that first?

Balarka Sen

The explicit topology is obtained from setting $V(\wp)$ to be the prime ideals containing $\wp$. These are the closed nbhds around $\wp$ in the Zariski topology on $\text{Spec} R$

user147690

@octatoan No you don't have to finish it, but it wouldn't hurt. Aluffi is a very nice book, but I haven't used it that much

octatoan

12:33 PM

Will Aluffi cover everything in a standard first linear algebra text?

Balarka Sen

You should do Artin, @octatoan, if you haven't done much of a linear algebra.

octatoan

I finally got around to "really" starting analysis, so I'm looking for another book to work through at the same time as baby Rudin.

Okay then.

brb checking Flipkart. I'm tired of semi-legal pdfs.

Balarka Sen

Artin provides good intuition for beginners. It also covers more topic than standard algebra texts, like, for example, doing algebraic number theory.

user147690

@octatoan How did you know it was a Unique factorization domain???

octatoan

:)

One picks things up.

Just happened to see the term around.

user147690

12:37 PM

@octatoan when I spelt the acronym wrong wut...

octatoan

Yeah, yeah, I know.

Balarka Sen

U DF? :P

user147690

Are you someones alt account :P

octatoan

No.

PIDs, UFDs . . . why?

Is Artin 2e the latest edition?

Balarka Sen

yes. well, that's what I have with me.

Chris's sis

12:38 PM

BBL (I wanna write up the proof)

octatoan

Mm. Flipkart?

Balarka Sen

?

octatoan

Where'd you get it from?

Balarka Sen

You mean if I bought it from flipkart?

octatoan

Yes.

I figure you buy a lot of books, so...

Balarka Sen

12:40 PM

As a matter of fact, I got it from college street. I rarely order books from the net.

user147690

Is this a joke.. You can buy D&F for $11AUD on flipkart?

octatoan

Economy editions!

Mwahahaha.

user147690

It's $100+ here...

Balarka Sen

Probably you'll be sent my copy of D-F.

octatoan

Sometimes the paper is crap though.

Balarka Sen

12:41 PM

... which looks like dishrag.

octatoan

Or worse.

user147690

@BalarkaSen Who will? Octatoan?

Balarka Sen

No, I mean if you try to buy the $11-copy.

user147690

Oh okay haha

user147690

Nah I'll buy beast hardcopies when I start TA'ing

octatoan

12:43 PM

@BalarkaSen any stores on College St you know are good? My dad'll be in the area today.

user147690

@octatoan today... wut... Isn't it like 7pm almost?

octatoan

I suppose the bigger shops are open in the evening.

user147690

Are you really who you say you are :P

octatoan

Yes, indeed I am.

user147690

Okay haha

octatoan

12:47 PM

Just always been a lazy bum. My dad always gets me whatever I need from there; I'd be damned if I knew the first thing about when the shops close.

Balarka Sen

@octatoan Sarat Books got some nice ones.

You should go there and see for yourself.

octatoan

OK, thanks.

Balarka Sen

@AlexClark How much do they pay you people for TA'ing?

user147690

@BalarkaSen They will pay me 44 per hour + 44 a week as a base for prep

Balarka Sen

ah.

user147690

12:51 PM

So I'll get atleast 6 hours a week

user147690

per class

user147690

So 308AUD(15600rp) a week per class I TA, I might TA two

Balarka Sen

You don't get to start TA'ing until you get to senior undegrad, though, right?

user147690

@BalarkaSen Yep - I'll TA discrete math 1 and calc 1 probably

Balarka Sen

From when?

user147690

12:53 PM

Next sem

Balarka Sen

oh, cool.

user147690

So July

user147690

Not locked in yet, but it's probable

Balarka Sen

what else can you TA?

user147690

@BalarkaSen Algebra I, but it's only in SEM 1, so I'll have to wait until next year, and I will TA Calc II after I have TA'd Calc I

user147690

12:54 PM

Next year hopefully I will TA this algebra course I am doing now :)

Balarka Sen

that'd be nice

user147690

(since I will have done 3 more alg courses and a research project by then)

Balarka Sen

what kind of research project have you done/will be doing?

user147690

@BalarkaSen Don't have any idea haha, I will take research next year first sem though for sure(since I need to fill my units up to graduate)

Balarka Sen

right. well, I am sure you'll have fun.

they really expect something like a survey when they mean research project, right?

user147690

12:56 PM

I will, and I will get a taste of what honours year will be like

user147690

@BalarkaSen Survey as in exposition? It is acceptable yes

user147690

But usually they give us the topic they are pretty much sure we can get something new from(it is a pretty high ranked uni)

user147690

smp.uq.edu.au/smp-research-projects

Balarka Sen

ah, I see.

robjohn

@Chris'ssis taking a look...

Balarka Sen

1:02 PM

ok, gotta run.

user147690

@BalarkaSen Cya later

Chris's sis

@robjohn OK :-)

robjohn

@Chris'ssis I believe they need that $p$ be even. If $p$ is odd, the sum is $0$, is it not?

Chris's sis

@robjohn Yeah, but it is even. It's $2p$.

robjohn

@Chris'ssis I see that now. Somehow on my first reading, I saw $p$

Chris's sis

1:14 PM

@robjohn Ah, OK. On that page the characters are petty small. I couldn't find the page from AMM.

robjohn

@Chris'ssis Off to the park, BBL

Chris's sis

@robjohn I'm off too for a while.

BBL

octatoan

1:51 PM

For when you guys come back: how much time does it take to finish baby Rudin, assuming you do the exercises?

user147690

@octatoan A bloody long time

user147690

@octatoan If by finish you mean every page fully read every exercise fully done in the entire book

octatoan

Is it necessary?

I really just want to get

user147690

@octatoan I doubt it

octatoan

a strong enough foundation that I won't have trouble later.

I won't do all the exercises!

user147690

1:55 PM

@octatoan In fact PMA has a few terrible last chapters I have heard

octatoan

Measure theory?

user147690

@octatoan Inclusive of that yep

user147690

But I think it might have been 8 and on are bad

octatoan

So how much should I do?

user147690

First 7 chapters would be nice

octatoan

1:57 PM

I just want to get started studying topology and algebra as quickly as possible.

user147690

@octatoan Well Analysis is quite nice as well

user147690

@octatoan I don't think you should rush analysis, it takes awhile to really appreciate

octatoan

I can do algebra simultaneously, can't I?

user147690

@octatoan Sure, but I think analysis is great for mathematical maturity

octatoan

Hmm.

So I'll be doing Artin and Rudin simultaneously for the summer break. gulp

user147690

1:59 PM

Sounds like fun

octatoan

Yeah.

user147690

It really does actually haha, I can't wait to have some time to myself to really give topology a shot

octatoan

Topology. Prerequisites?

user147690

@octatoan Analysis in my opinion

octatoan

OK.

user147690

2:00 PM

@octatoan Really there aren't any, but rudin does basic topology in chapter 2 of PMA, and I think that is really important

octatoan

Later. :)

Thanks.

user147690

@octatoan Cya

Paul Plummer

This is the Ghost of Paul, here to exact my revenge @AlexClark

user147690

@PaulPlummer NOOOOOOO

Paul Plummer

this and this are pretty cool @AlexClark

user147690

2:10 PM

Both are worrying and both are the same?

octatoan

Oh, and I forgot to get an estimate. How much can I reasonably hope to cover in a month @AlexClark?

user147690

@octatoan Hmmmm

user147690

@octatoan How old are you or what grade are you in>

octatoan

Does that matter?

user147690

For most people yes, for balarka no

octatoan

2:11 PM

I've done calculus at the level of Spivak.

How old is he?

user147690

I think 16

octatoan

Tenth or so, I think?

Yup, same here.

Paul Plummer

Apparently my copy paste fuu is no good here

octatoan

But, yeah, I see what you mean.

About him.

user147690

He is graduate level lol

octatoan

2:13 PM

I know. Anyway, please . . . estimate?

Paul Plummer

There is no point in worrying about how long it will take, it will take as long as needed... Its not like you are on a dead line

user147690

Just looked through it, and honestly I have no idea. It would vary significantly

octatoan

I'll just try to do as much as I can comfortably cover then, I guess.

user147690

If you have done Spivak properly than you should already know chapter 1,3,4,5 right?

octatoan

Yes.

user147690

2:15 PM

2 will take you a week(if you do very well)

octatoan

I need to brush up on my epsilondeltamanship.

It's been a while

user147690

And 1,3,4,5 will be done very differently and much more rigorously I imagine

octatoan

Hm, yes.

user147690

If you do amazingly the entire time, you might get up to half way through chapter 6

user147690

Only assuming you really do know spivak well on top of doing amazingly

octatoan

2:16 PM

Construction of $\mathbb{R}$ from $\mathbb{Q}$ is new for me.

user147690

Honestly skip the construction until afterward in my opinion

octatoan

Yeah, that's a sort of target then.

Okay.

user147690

Are you talking about the appendix construction using several steps if so yeah, that isn't worth the time(in my opinion, maybe others would suggest to do it)

octatoan

Yes, the appendix

I love that word, haha.

BBL then.

user147690

ta ta as balarka says

Paul Plummer

2:20 PM

see yah

user147690

I'm not leaving if that was to me by mistake

user147690

It's too early

Paul Plummer

It was to you, but I was going to play it off as if it was to octatoan (sort of like the whole trip and start running thing)

user147690

Hahahaha fair enough

octatoan

Balarka speaks in Bangla here as well?

user147690

2:22 PM

Wait what is 'ta ta' not a British thing...?

Paul Plummer

Maybe it was borrowed while the British were imperialising (not sure from who)

user147690

Oh wow never considered that

Paul Plummer

Although it does seem to be a brit thing, at least wiktionary does not say anything about India.... probably not the best source

user147690

This wordpress suggests that it is british origin, but I am too tired to read it

user147690

Since its 12:30am

Paul Plummer

2:28 PM

Which is to early...

Balarka Sen

"Ta ta" is indeed a British thing.

user147690

@PaulPlummer Too early to stop doing math, not too early to stop reading non-math

octatoan

I did not know that, TIL

Balarka Sen

Interesting point by @Paul. I don't know if we used to say that before they came here.

Hmm, I don't think I have ever read 'ta ta' in any of the pre-imperial or imperial age books I know of.

user147690

Earliest use is 1837 so it is impossible to know

Paul Plummer

2:32 PM

Oh, found a comment online that seems to suggest it came from the brits, but it is so ingrained in some parts of india, that many don't know it is "english", Although I am personally not conviced it is actually English, they just did a good job of convincing people they did it...

Balarka Sen

@AlexClark wait, 1837 even existed?

user147690

@BalarkaSen Apparently the earliest we have documented evidence

Rammus

I'd say tiggr is to blame for its association with english "TTFN: ta ta for now"

user147690

Wait previous word for it was da da apparently

Balarka Sen

lol

octatoan

2:36 PM

That's something totally different @AlexClark

user147690

early 19th century: of unknown origin; compare with earlier da-da .

octatoan

Oh, in English?

Oh ok.

Paul Plummer

This would probably be a good question for "English" stack exchange (or maybe "India" stack exchange if there is one)

octatoan

There isn't

Balarka Sen

Maybe something like a history stackexchange?

user147690

2:37 PM

Why is this bugging me so much lmao

octatoan

I can now check off "Turn a math chatroom into a linguistics one" off my list of things to do before I die.

user147690

Hahahahaha

Balarka Sen

I am going to give up doing research on linguistics now.

user147690

Same

Paul Plummer

Haha, at some point I might ask for a definitive answer on one of the linguist type stack exchanges, but not right now

user147690

2:38 PM

@PaulPlummer I would be keen to see that hahahaha

iwriteonbananas

@BalarkaSen let $X= X_1 \sqcup X_2$. how do i show that the inclusions $X_i \to X$ induce an isomorphism $H_n(X_1) \oplus H_n(X_2) \to H_n(X)$ ?

surely, we need long exact sequence

Balarka Sen

nah.

construct an inverse.

iwriteonbananas

how?

Balarka Sen

construct an inverse at the chain level first.

$\sigma : \Delta^n \to X$ be a singular simplex. domain is connected, thus so must be the codomain. so $\im \sigma$ goes inside some $X_i$.

this gives you an easy map $H_n(X) \to H_n(X_1) \oplus H_n(X_1)$ after modding out by boundaries.

verify that it's an inverse of that map in $\mathbf{Grp}$

iwriteonbananas

ok, i understand but this is a general homology theory (with values R-modules), not necessarily singular homology

Balarka Sen

2:47 PM

i am not gonna edit that again

@iwriteonbananas oh.

you should have mentioned that before :P

iwriteonbananas

you're right, sry

Balarka Sen

but then it's almost a tautology : it's that one of the axioms of homology?

iwriteonbananas

no, it's not an axiom of homology

the axioms are homotopy invariance, long exact sequence and excision

Balarka Sen

oh, i guess i have forgotten what a homology theory is.

right, then you need to use the long exact sequence

mayer vietoris, i.e.

iwriteonbananas

yeah, that's the only chance

oh, yes mayer vietoris sounds promising

but for mayer vietoris, we need sets whose interiors cover $X$

wait that's singular homology

Balarka Sen

2:59 PM

@iwriteonbananas interiors needn't cover X in M-V sequence for arbitrary homology theories?

i don't think that's true.

but anyway, connected components do cover the whole space

Transcript for

$Mathematics$ Mathematics