Mathematics - 2015-03-13 (page 1 of 5)

iwriteonpeoplewhowriteonbanana

00:00

@Owatch Can you apply the formula tu turn some exps to sins now ?

ᴇʏᴇs

Right, but in my notes it says $-nz_0^{n-1}$ becomes $-z_0^{n-1}$ but I don't see why

Owatch

No I can't. I can make $e^{3ix} - e^{-3ix} = 3sin(ax)$, but I don't know if its right, I don't know where to put it.

And remove 2i from denonimator

iwriteonpeoplewhowriteonbanana

@ᴇʏᴇs it's a typo I believe. Check for $n=2$

@Owatch So we get $\frac{e^{3ix}-e^{-3ix}}{4*2i}-3\frac{e^{ix}-e^{-ix}}{4*2i}$

ᴇʏᴇs

@iwriteonpeoplewhowriteonbanana I don't think it's a typo, in my notes I literally wrote "$nz_0^{n-1}$ becomes $-z_0^{n-1}$ because..." and I didn't finish writing because the professor erased the board before I got to write it

iwriteonpeoplewhowriteonbanana

@ᴇʏᴇs check for $n=2$

@Owatch right ?

OOH nvm

Soo we have $-\frac{e^{3ix}-e^{-3ix}}{4*2i}+3\frac{e^{ix}-e^{-ix}}{4*2i}$

Saal Hardali

00:05

Attention all complex analysts
Let $f: \mathbb{C} \to \mathbb{C}$ be an entire function. What can generally be said about curves in $\mathbb{C}$ upon which $arg(f)$ is constant (where the function doesn't change phase).

Do they always exist? Do they have special properties beyond the continuity that we impose?

iwriteonpeoplewhowriteonbanana

right ?

Owatch

How does that give you 8i^3 on bottom?

Where did you get 3 from?

iwriteonpeoplewhowriteonbanana

it doesn't. There's a factor 1/i missing (in my expression)

Saal Hardali

Plus could someone help here?
http://math.stackexchange.com/questions/1183864/generalization-of-the-argument-principle

Owatch

Well why is it missing, and why is there a +3 on the second piece?

iwriteonpeoplewhowriteonbanana

00:07

@Owatch 3 = 2 + 1 (check the factors involved)

Owatch

I don't get it.

ᴇʏᴇs

@iwriteonpeoplewhowriteonbanana Do you have a website or book for a proof that if $\sum\limits_{n=0}^{\infty} a_n z^n$ has a radius of convergence $R>0$, then $f(x) = \sum\limits_{n=0}^{\infty}$ is holomorphic on the open disk $D(0,R)$, and for all $z \in D(0,R)$, $f'(z) = \sum\limits_{n=1}^{\infty} na_z z^{n-1}$

iwriteonpeoplewhowriteonbanana

$\frac{e^{3ix}-2e^{ix}+e^{-ix}-e^{ix}+2e^{-ix}-e^{-3ix}}{8i^{3}}=\frac{e^{3ix}-e‌^{-3ix}} {8i^3}-3\frac{e^{ix}-e^{-ix}} {8i^3}$

@Owatch ^

@ᴇʏᴇs The second part is clear. I don't remember what a holo function is though, mind reminding me ?

Owatch

Okay.

iwriteonpeoplewhowriteonbanana

@Owatch And $i^3=-i$

Thus it's equal to $\frac{1}4(3\sin(x)-sin(3x))$

Owatch

00:11

(2+1)(e^ix - e^-ix) gives everything except the cubed e's.

ᴇʏᴇs

@iwriteonpeoplewhowriteonbanana A complex-valued function on an open subset of the complex plane is holomorphic (or analytic) if its derivative exists for each element in the subset

Owatch

@iwriteonpeoplewhowriteonbanana I do not know this either.

iwriteonpeoplewhowriteonbanana

@Owatch You don't know complex numbers ?

Owatch

I have never seen anything used to simplify this problem. I will fail if I see it again.

I don't know complex numbers.

ᴇʏᴇs

@iwriteonpeoplewhowriteonbanana In the proof of this is where the professor turned the $nz_0^{n-1}$ into $z_0^{n-1}$

iwriteonpeoplewhowriteonbanana

00:12

@Owatch Well then learn $sin^3(x)=\frac{1}4(3\sin(x)-sin(3x))$

ᴇʏᴇs

Otherwise it doesn't seem like the proof will work

iwriteonpeoplewhowriteonbanana

And do the integral with that @Owatch

@ᴇʏᴇs Have you checked the case n=2 ? Does your formula work ?

ᴇʏᴇs

@iwriteonpeoplewhowriteonbanana No, which is why I need to compare it to a correct proof but none of my complex analysis books have this proof

iwriteonpeoplewhowriteonbanana

@ᴇʏᴇs I might have it, but only on R. Let me check.

ᴇʏᴇs

Thanks

user159870

00:16

@iwriteonpeoplewhowriteonbanana It's for a friend of mine, I don't know what definition he uses. I asked him but I don't have an answer yet. How do we get the telescopic series???

ᴇʏᴇs

I wouldn't care so much about the proof as the result, but the prof. says he's gonna ask this proof on the exam

Owatch

$\frac{1}{4}\int 3xsinx - xsin3x dx$

iwriteonpeoplewhowriteonbanana

@user159870 Well the definition is like $\sum_\Bbb{m\in\Bbb{Z}} x(m)h(n-m)$

Owatch

From $\int xsin^{3}xdx$

iwriteonpeoplewhowriteonbanana

@ᴇʏᴇs Do you know how to do it on $\Bbb{R}$ ?

It's pretty straighforward

@Owatch Indeed

ᴇʏᴇs

00:19

@iwriteonpeoplewhowriteonbanana No, I didn't know what a power series is until today

iwriteonpeoplewhowriteonbanana

@ᴇʏᴇs Ok so 1st quick lemma : $\sum a_nx^n,\sum na_nz^{n-1}$ have the same ROC

ᴇʏᴇs

@iwriteonpeoplewhowriteonbanana Really? This proof took up 2 pages for me (maybe I just write big)

Owatch

$\frac{1}{4} \int 3xsinx dx - \frac{1}{4}\int xsin3x dx$

iwriteonpeoplewhowriteonbanana

Then swap the order of integration @ᴇʏᴇs

done

ᴇʏᴇs

@iwriteonpeoplewhowriteonbanana We needed 2 additional lemma including that one

Owatch

00:21

These look like two cases of integration by parts?

user159870

@iwriteonpeoplewhowriteonbanana How do we find $\sum_{m \in \mathbb{Z}} (u(m)-u(m-T))e^{-a(n-m)}u(n-m) $ ?

iwriteonpeoplewhowriteonbanana

@user159870 Unless we have mire information on $u$, I don't know. I was just hoping that the $u_n$ would telescope.

@Owatch I guess

@ᴇʏᴇs Which ones ? I don't see why you'd need that

ᴇʏᴇs

@iwriteonpeoplewhowriteonbanana Lemma 1: Given a power series, there exists an $R$, $0\leq R \leq \infty$ such that $\sum\limits_{n=0}^{\infty} \lvert a_n\rvert \lvert z \rvert ^n$ converges if $\lvert z \rvert < R$ and $\limsup_{n \to \infty} \lvert a_n \rvert \lvert z \rvert ^n = \infty$ if $\lvert z \rvert > R$, and Lemma 2: If $a, b \in \mathbb{C}$, $\lvert a \rvert < \rho$, $\lvert b \rvert < \rho$, then $\lvert b^k - a^k \rvert \leq k\rho^{k-1} \lvert b - a \rvert$ for all $k \geq 0$

iwriteonpeoplewhowriteonbanana

but lemma 1 is obvious

except maybe for the lim sup part, which is Hadamar's stuff. I never needed lim sup.

ᴇʏᴇs

And then we used some corollary of that

iwriteonpeoplewhowriteonbanana

00:28

I find my way easier q_q

Louis

@infinitesimal For omega < 1 I see the triangle having greater 180 degrees. When trying to concentrate on the gridlines as a way to see a side curved inwards, I fail to not see them bulging outwards... so I was wondering if there is an error in that image, or if it' fine and I'm just seeing it the way I do which is wrong.

iwriteonpeoplewhowriteonbanana

Just swap the integration-derivative

ᴇʏᴇs

If $r < R$, then the convergence is uniform on the set $\lvert z \rvert \leq r$ and if $z \in \mathbb{C}$ and $\lvert z \rvert < R$, then $\sum\limits_{n=0}^{\infty} a_n z^n$ converges and if $r < R$ then the convergence is uniform on $\lvert z \rvert \leq r$

Owatch

$\frac{1}{4}(-3xcosx+3sinx) - \frac{1}{4}(\frac{-xcos3x}{3}+\frac{sin3x}{9}) + c$

ᴇʏᴇs

@iwriteonpeoplewhowriteonbanana We don't have integrals yet

iwriteonpeoplewhowriteonbanana

00:30

Ah.

Semiclassical

can anyone else make sense of what this question is asking?

iwriteonpeoplewhowriteonbanana

@ᴇʏᴇs gowers.wordpress.com/2014/02/22/differentiating-power-series

ᴇʏᴇs

Thanks @iwriteonpeoplewhowriteonbanana

Mike Miller

@Semiclassical i doubt it would be a good question even if i could

Conor O'Brien

G'night, all!

ᴇʏᴇs

00:35

@iwriteonpeoplewhowriteonbanana Oh, that uses differentiation in $\mathbb{R}$ which I don't think we can use in our complex proof

Semiclassical

agreed

iwriteonpeoplewhowriteonbanana

It's awfully late here, i'm off to sleep. Good night !

ᴇʏᴇs

Night @iwriteonpeoplewhowriteonbanana

iwriteonpeoplewhowriteonbanana

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ᴇʏᴇs

Thanks for helping

iwriteonpeoplewhowriteonbanana

00:35

@ᴇʏᴇs Good luck finding a nice proof :-)

Committing to a challenge

Two matrices, $A$, $B$, are similar if some invertible $P$ exists, such that $P^{-1} A P = B$. Does it matter if it were $PAP^{-1}=B$ instead? I think not, but I wonder if it matters when trying to show that similarity is a reflexive relation

$P^{-1} A P = A\implies PAP^{-1} = A \implies P=P^{-1} \implies$ $A $ is similar to $A$

Owatch

Solved I think.

Committing to a challenge

Neat handwriting

user159870

@robjohn @iwriteonpeoplewhowriteonbanana The denition is the infinite sum x(n-k)*h(n)

ᴇʏᴇs

@Owatch Do you write slowly

Owatch

00:52

I work carefully.

Committing to a challenge

I write so damn fast that noone else can read it :)

It's an auto cipher <3

ᴇʏᴇs

I write pretty fast that even I can't read it

Committing to a challenge

Encrpytion by converting letters into squiggles

The decryption method is me reading it out

ᴇʏᴇs

Yea my handwriting is just horizontal lines of varying heights

Committing to a challenge

ROFL

Owatch

00:53

I often miss signs, ect. And as I solve I write. Since I solve slowly, I have time to print well.

A Beautiful Mind

Hi, I am still alive.

ᴇʏᴇs

Hi @ABeautifulMind

Good to know

Owatch

I'm tired. I think I do one more problem. Then go do other stuff.

ᴇʏᴇs

@Owatch Other stuff = different math subjects?

Owatch

I must find a man called James Stewart.

He is on the front of this book, I have to stop him.

Committing to a challenge

00:56

@ABeautifulMind We are all glad for that.

Owatch

No, other stuff does not include math subject.

A Beautiful Mind

I will be in this chat for one hour, after that I will take a nap.

Committing to a challenge

For exactly one hour?

ᴇʏᴇs

@Owatch Did you know he has a mansion worth millions of dollars

Owatch

James Stewart has taken away my vacation.

I want to make him pay.

Or anyone else who was involved in writing this book.

A Beautiful Mind

00:59

@Committingtoachallenge Nope, but approx.

Committing to a challenge

@ABeautifulMind What have you done today?

A Beautiful Mind

@Committingtoachallenge I have been in a state of great anxiety the entire year, just thinking about the past, present and future mostly.

Committing to a challenge

Lying in bed thinking about these things?

ᴇʏᴇs

@ABeautifulMind I hope you think about positive things

A Beautiful Mind

@Committingtoachallenge Or in other positions.

Committing to a challenge

01:01

You sit there thinking to yourself in your head?

A Beautiful Mind

@Committingtoachallenge Or talking aloud.

@ᴇʏᴇs Yes, I am trying to do that.

Owatch

Think about absolute values. They are always positive.

Committing to a challenge

Talking aloud improves my reasoning abilities

A Beautiful Mind

I know some people will tell me I should not think so much but should do other things, but this is something I have to go through. If you understand what happened in 33 years of my life, you will understand everything.

ᴇʏᴇs

I like to walk while I'm thinking which isn't good for my exam-taking skills

A Beautiful Mind

01:03

I am trying to make progress, of course, as always.

robjohn

@user159870 usually, it is $\sum\limits_{k}x(n-k)h(k)$

A Beautiful Mind

@Committingtoachallenge and @ᴇʏᴇs are my good friends.

Owatch

$\int sin^{3}x cos^{4}x dx$

I convert stuff

$\int sin^{2}xsinx(cos^{2}x)^2dx$

Committing to a challenge

Glad to hear it :). When you start doing Math again, perhaps we can work on a paper :)

A Beautiful Mind

Honestly I am not even sure if I will be well again, but it is something I hope for every day.

I call something hard to achieve but possible a miracle, and I am trying to achieve this miracle.

@Committingtoachallenge By then you would have won the Fields medal.

Mike Miller

01:07

@robjohn I am free 10-12 tomorrow. If you are not busy then, maybe that is when we could all meet.

Committing to a challenge

Haha that would be a miracle

I think meeting two other M.SE members simultaneously would make me anxious as hell[in the nervous sense]

A Beautiful Mind

Do you have social anxiety? I don't.

Committing to a challenge

Yes I do - but I don't take medication for it[I did a long time ago{beta blockers}]

A Beautiful Mind

I imagine @ᴇʏᴇs and @Committingtoachallenge are very good looking.

Committing to a challenge

@ABeautifulMind Haha nah, I am somewhere in 7-9 depending on who you ask(or a 10 if you ask my more attractive than me girlfriend)

ᴇʏᴇs

01:09

@ABeautifulMind I'm ugly as heck

A Beautiful Mind

One day we will all meet up.

ᴇʏᴇs

Even my mom tells me so

Blessoul

Hello

ᴇʏᴇs

Hi @Blessoul

How are you

A Beautiful Mind

@ᴇʏᴇs Your mom is crazy as we all know.

Committing to a challenge

01:10

wut? That is depressing @EYESZ

Blessoul

Hi @ᴇʏᴇs

Doing well, going crazy over my advanced combinatorics class.

what about you guys?

A Beautiful Mind

You have not become crazy until you reach my stage.

Committing to a challenge

Doing my algebra assignment

(still)

Blessoul

lol, do you happen to know combinatorics and might be able to help me with a problem?

Committing to a challenge

I don't happen to know combinatorics(sorry)

Blessoul

01:12

Oh man

no need to be sorry

Committing to a challenge

Do you happen to know abstract algebra?

Blessoul

whats the problem

ᴇʏᴇs

Hi @Ted

Ted Shifrin

hi mr eyeglasses

A Beautiful Mind

Hi Ted, he is Bart.

Ted Shifrin

01:14

I don't think we should put too much faith in your book-burning mother.

A Beautiful Mind

@ᴇʏᴇs is Bart and @Committingtoachallenge is Alex. And you know who I am @ted.

Ted Shifrin

I like the name mr eyeglasses :P

Mike Miller

eyebrows

Ted Shifrin

@Mike, don't you have a lecture to prepare?

ᴇʏᴇs

lol

Mike Miller

01:15

Doing so

Now that I have the right version of this paper...

Ted Shifrin

Did anyone answer your question on MO? And is Pedro in the vicinity yet?

oh, there were multiple versions?

Committing to a challenge

Let $k$ and $N$ be positive integers and suppose $k\leq N$. Let $W(k)$ denote the vector space spanned by vectors $V_S$ where $S$ ranges over subsets of $\{1,\dots, N\}$ of size $k$.

For $S\subset \{1,\dots,N\}$, let $S^c$ denote the complement. Show that the map sending $V_S$ to $V_{S^c}$ defines an isomorhism of vector spaces $W(k) \to W(N-k)$

Is what I am working on @Bless

Well its a subquestion but that's a closed form question

Good old first week assignments that are too hard

Mike Miller

Yes - Ruberman (one of the topology students here is going to do a postdoc with him) commented, saying the desired theorem is proved in section 22, and exposited it a bit. Now, the paper I was looking at was 4 pages long, and I didn't think you could fit 22 sections in it.

Ted Shifrin

Do you have a fixed set $V$ of $N$ vectors to start with, @Committing?

Mary Star

@TedShifrin Could you take a look at math.stackexchange.com/questions/1187196/…? ??

Mike Miller

01:16

Turns out I was reading a version published in the Bulletin...

Ted Shifrin

Ah, Ruberman's another grad school compatriot, @Mike ... actually heard from him recently.

Mike Miller

where he just announced the results, of course. I was upset that Gluck seemed to think the result was so trivial, since he wrote it in two lines!

thank the lord it's not ALL my fault :)

user159870

@robjohn $\sum\limits_{k}x(n-k)h(k)= \sum\limits_k (u(n-k+2)-u(n-k-2) \left( \frac{1}{2} \right)^{|n|}$
The Heaviside function gives the unit step signal(t).
Do do we continue with the sum???

A Beautiful Mind

My favourite line from my favourite movie is 'it is not your fault'.

Ted Shifrin

@MaryStar: I have no idea what the intersection refers to in your question. You have got to stop asking us every single one of your homework questions.

A Beautiful Mind

01:19

@TedShifrin She is a star just like me.

Committing to a challenge

342 questions holy moly @MAry

Ted Shifrin

Everything is always your fault, @Mike. :P

Mike Miller

Aw.

Committing to a challenge

@TedShifrin Nope

Ted Shifrin

Oh right, a few things are @Committing's fault.

Committing to a challenge

01:20

Noooo I was noing the previous thing follow the arrow haha

Don't make things my faultttt

Philip Hoskins

If M, N are smooth manifolds and F from M to N is a submersion, then every smooth vector field Y on N has a lift X on M. But why is it the case that if F is not a diffeomorphism, then this lift is not unique?

A Beautiful Mind

@Committingtoachallenge Holy Mary.

Ted Shifrin

Then it seems to me $W(k)$ is the entire vector space, if I can pick any $k$ vectors I want each time, @Committing. That problem sucks.

Committing to a challenge

@MaryStar You have 4 questions today!? Do you think about them before posting them?

A Beautiful Mind

@Committingtoachallenge youtube.com/watch?v=GtkST5-ZFHw here.

Ted Shifrin

01:22

Because you can add to $X(p)$ any vector in $\ker(dF_p)$, @Philip.

ᴇʏᴇs

@ABeautifulMind I cried

Mike Miller

@PhilipHoskins Try the most trivial example: $\Bbb R \to \{pt\}$. Then every vector field on $\Bbb R$ is a lift of the (only) (trivial) vector field on the point. This is prototypical of what's going on in general.

ᴇʏᴇs

RIP Robin Williams

Committing to a challenge

@TedShifrin I haven't been working on this one long, so maybe I don't fully understand it, I will show you my proof when I finish it(just so you can see if it was a bad question in the end)

after I hand in of course, I don't want to get in trouble haha

Mary Star

@Committingtoachallenge I ask something specific at each question. I have tried something and asked at the point I got stuck...

Ted Shifrin

01:23

Perhaps I don't understand the question, @Committing. That's why I asked if you had a fixed set of vectors to start with, but even then, if you can pick any $k$ of them, you will generate their entire span if you get to choose all possible $k$-tuples.

A Beautiful Mind

@ᴇʏᴇs Have you watched the whole movie?

Committing to a challenge

@Mary really? That's what you do?

A Beautiful Mind

@Committingtoachallenge Maybe she got stuck at the start, that's possible.

@MaryStar I admire your spirit. Are you planning on becoming a mathematician though?

Mary Star

Yes!! @ABeautifulMind

Ted Shifrin

Oh, I see, $V_S$ is chosen for a fixed $k$-tuple $S$. I'm still confused. You needn't choose linearly independent vectors. Agh.

Philip Hoskins

01:26

@TedShifrin Okay, that was pretty much what I thought. But, just to be clear, dF_p only makes sense locally where we can ensure $F$ is surjective, right? For example, if I take a very small coordinate cube of M, then F will map onto some small coordinate cube of N by the Rank Theorem.

A Beautiful Mind

@MaryStar OK. I think you will need to work very hard to achieve that, good luck.

Ted Shifrin

@Philip: No, if you have a map $F\colon M\to N$, $dF_p\colon T_pM\to T_{F(p)}N$ is defined independent of charts.

Karl Kronenfeld

@Committingtoachallenge You said it is a subquestion of something. By any chance are any of the things you use defined in that primary question?

Ted Shifrin

No hypothesis on $F$ other than smoothness.

Karl Kronenfeld

Hi @Ted @Mike

Ted Shifrin

01:27

heya @Karl

Mike Miller

morning karl

Committing to a challenge

@TedShifrin Assumed linearly independent is he said in class briefly(without updating the assignment)

A Beautiful Mind

Whenever Karl comes, I get confused with Arkamis and Ed.

Philip Hoskins

@Ted: But You don't know that dFp(X) will be a vector field on N.

Owatch

I think I solve.

Easy

Karl Kronenfeld

01:28

@MikeMiller You figured me out.

Owatch

Ted Shifrin

LOL, @Committing ... oh, isn't he helpful :P

A Beautiful Mind

@Owatch Very nice handwriting.

Committing to a challenge

@KarlKronenfeld The only other subquestion prior was find dimension of W(k)

Ted Shifrin

@Philip: I was assuming you had defined a smooth lift $X$ and was explaining why it needn't be unique.

Mike Miller

01:29

@Karl: Are you willing to tell me if you're an undergrad or not?

Owatch

thankyou

Karl Kronenfeld

@MikeMiller Yeah, I am an undergrad.

Mike Miller

Willing to say what year, or am I pushing it?

Karl Kronenfeld

First year

Mike Miller

whoaly moly

Ted Shifrin

01:29

@Owatch: Neatness really does help in mathematics. I once had a student who got a D from me in one course ... he couldn't even read his own work. I forced him to be neater, and he improved to a C in the next course (which was harder). I think I have the grades right.

A Beautiful Mind

I thought you finished undergrad long ago @KarlKronenfeld? Maybe I forgot.

Philip Hoskins

@TedShifrin Yes, but to show the existence of the lift, I needed to work locally. So, if I want to make a vector field that gets killed by dF, I'm guessing I would have to also work locally. Is there a way to construct the lift by starting globally?

Owatch

Thank you for inspiration.

Committing to a challenge

@KarlKronenfeld Lies haha

Or trick answer

Mike Miller

@Committingtoachallenge Well, there's no reason to believe he's not.

A Beautiful Mind

01:30

@TedShifrin Better handwriting = better grades. Now all I need is to hand in a blank sheet and I get an A+.

Committing to a challenge

@MikeMiller Except he has been on M.SE for 2 years has has 86 points in abstract algebra

ᴇʏᴇs

@ABeautifulMind Yes I watched the whole film

Karl Kronenfeld

@Committingtoachallenge Because I am concerned with people's perceptions of first-year undergrads in relation to me. /sarcasm

Ted Shifrin

@Philip: You're right that you need to work locally to get a lift. I don't know how fancy you are, but because $F$ is smooth and has constant rank, $\ker dF$ defines a vector subbundle of $TM$ and you can take any smooth section of it.

ᴇʏᴇs

@ABeautifulMind Would you defend me if I was making a fool out of myself like in the bar scene

Ted Shifrin

01:31

Jasper, I'll let you look for the fallacy ...

A Beautiful Mind

@ᴇʏᴇs If I could, I would.

Karl Kronenfeld

@Committingtoachallenge Oh I think I get it. Still a shitty question, if so.

Ted Shifrin

@Karl: It seems we all thought you were a graduate student at this point, despite your coy behavior :P

Committing to a challenge

The guy is pretty impressive(the lecturer) surely he wouldn't give shit questions right

Mike Miller

@Ted: My topology professor couldn't read my handwriting. He told me to give him something legible or he wouldn't read it, and would just give it a 0. He suggested LaTeX. And that's the story of how I learned LaTeX.

A Beautiful Mind

01:32

And you guys know all about me, no need for more info.

Karl Kronenfeld

@TedShifrin My childishness should have set you all straight.

Mike Miller

Ted and I are perhaps more childish than ye.

Owatch

coy: (especially with reference to a woman) making a pretense of shyness or modesty that is intended to be alluring.

Committing to a challenge

@KarlKronenfeld You are listed as 22?

Owatch

He said you are a woman karl.

Karl Kronenfeld

01:33

ok, enough questions.

Ted Shifrin

don't be sexist, @Owatch: Men can be plenty coy.

Owatch

What are you going to do about that?

A Beautiful Mind

@MikeMiller That's very nasty of him.

Mike Miller

No it's not.

ᴇʏᴇs

I started using LaTeX in community college because I thought it looked cool

Committing to a challenge

01:33

Lot's of chat

Owatch

@TedShifrin It's definition, I didn't write.

Ted Shifrin

Your dictionary is old-fashioned and sexist :P

ᴇʏᴇs

Handed in all my homeworks in LaTeX and felt cool because everyone was jealous of how pretty it looked

A Beautiful Mind

You have used LaTeX, but have you used latex? LOL.

ᴇʏᴇs

@ABeautifulMind pls

@ABeautifulMind I wear latex gloves whenever I eat pizza because I don't want to get oil on my hands

Ted Shifrin

01:34

mr eyeglasses, please smack him.

on second thought ...

ᴇʏᴇs

lol

Owatch

Ted Shifrin

Get back to your integrals, @Owatch.

A Beautiful Mind

Please don't flag me, I have been suspended too many times. 6 already, lol.

Ted Shifrin

There's no call for flagging.

Owatch

01:35

Yes Herr Feldmarschall Ted.

Ted Shifrin

As one many of whose relatives died in the concentration camps, @Owatch, I'm not sure I appreciate your humor.

Mike Miller

@Abe It's not against the rules to make jokes that aren't funny, or they'd have kicked me out long ago.

2

ᴇʏᴇs

I want to strengthen my differentiation/integration skills so I can take differential geometry in 2 years

Philip Hoskins

@TedShifrin Ok I had to think about that for a minute but it makes sense now. This is all still a foreign language for me.

A Beautiful Mind

But let's say that this room is very flag prone, lol.

Philip Hoskins

01:36

Thanks for the help.

Ted Shifrin

mr eyeglasses, it's your multivariable calculus and linear algebra skills you need ... and, depending on who teaches it at what level, differential equations occasionally or more advanced multivariable analysis.

Mike Miller

@Phil: Also, note that your statement wasn't quite right. We just need it to be a local diffeomorphism to have a unique lift.

Ted Shifrin

@Philip: Of course, to get smooth sections of vector bundles, I suppose we have, abstractly, to take local ones (coming from local triviality) and glue them together with partitions of unity. So you really win the argument :P

Owatch

Feldmarschall is not a position that implies involvement in WW2 tragedies.

But I apologize.

ᴇʏᴇs

@TedShifrin The chair who also taught the course says I should be comfortable with tensor and exterior algebra, change of variables formulas for multiple integrals, and existence and uniqueness theorems for ODEs

Ted Shifrin

01:38

Yes, @Owatch, I know what a field marshall is ... However, the German tone ...

Philip Hoskins

@MikeMiller Oh that's right, as long as dF_p is invertible in some neighborhood of each point, it'll be unique.

Owatch

Implies strictness

Ted Shifrin

mr eyeglasses, that sounds like a graduate course in manifolds, not an introductory undergraduate differential geometry class. For the graduate course, you absolutely need the inverse function theorem and multivariable analysis.

Philip Hoskins

@Mike I mean dF_p is invertible for all p in some chart. Sorry

Mike Miller

No need to apologize. I thought what you originally said was perfectly clear.

Ted Shifrin

01:39

@Philip: Again, the linear map $dF_p$ does not need a chart to define. :)

mr eyeglasses, who is the chair to whom you refer? I'm curious.

Mike Miller

He's actually just a large chair

He gets very uncomfortable when students sit on him

as do we all.

ᴇʏᴇs

@TedShifrin Do you have any recommended texts for tensor/vector analysis, etc. that I can read over the next couple years to prep me for this course

Ted Shifrin

smacks @Mike

ᴇʏᴇs

@Ted Józef Dodziuk

Ted Shifrin

mr eyeglasses, no, I would recommend you find an undergraduate differential geometry course first and then take a multivariable analysis course and point-set topology.

Ah, I know him.

ᴇʏᴇs

01:42

Actually I don't think he's the chair anymore

Ted Shifrin

He was a postdoc at MIT when I was an undergrad there ... so he's getting old :P (I think I have that right.)

ᴇʏᴇs

Yea, he was a Moore Instructor

A Beautiful Mind

Not Moron Instructor?

Philip Hoskins

@TedShifrin That's clear. I was only referring to it locally because that's where I was working to get that the lift will be unique when F is a local diffeo.

Mike Miller

@Ted: In lots of old topology papers, a (codimension $k$) smooth embedding of $S$ is defined as one that can be thickened to a neighborhood $S \times D^k$. But what we now call smoothly embedded surfaces in 3-manifolds can be 1-sided just fine. Funny how terminology evolves.

Ted Shifrin

01:43

right, @Philip ... You've got it :)

I'm confused, @Mike. If you have a non-orientable surface, you can still take a tubular neighborhood.

Mike Miller

You're right. Let me edit to the correct statement.

Ted Shifrin

Well, the original definition agrees with mine for what the local structure of an embedding is. I guess I still don't understand your 1-sided option. How do you have a one-sided tubular neighborhood unless you're talking about the boundary?

ᴇʏᴇs

@Ted Your multivariate calc text isn't enough analysis for the graduate differential geometry course is it?

A Beautiful Mind

The people in math chat are much nicer than those in Eng chat.

Mike Miller

@Ted If you embed something with nontrivial normal bundle.

Ted Shifrin

01:46

mr eyeglasses, yeah, it probably is. You still need tensors, though, and more topology.

ᴇʏᴇs

@Ted I am taking a topology course next year so that's set

Ted Shifrin

OK, so let's look at $\Bbb RP^2$ in $\Bbb RP^3$, @Mike.

Philip Hoskins

Ok cool. Thanks very much @Ted and @Mike

Ted Shifrin

Good to see you again, @Philip.

The tubular neighborhood theorem still gives you a neighborhood of the zero-section as diffeo to a neighborhood in the ambient manifold.

Mike Miller

@Ted: That has nontrivial normal bundle, and the tubular neighborhood is homeomorphic to the (open) unit disc bundle. We delete our copy of $\Bbb{RP}^2$ in this and we get something connected.

Ted Shifrin

01:48

Oh, I see, your $S\times D^k$ was a global product structure, so, yes, that happens only for trivial normal bundle. So I don't agree that that's the definition of embedding.

Mike Miller

Not anymore, no. :P

Ted Shifrin

Embedding meant trivial normal bundle? I don't recall ever seeing that.

Was that back when the dinosaurs roamed the manifold earth?

Mike Miller

Well, not that old - this is in a paper from '62.

Ted Shifrin

Maybe they were doing something with tame embeddings or something

In the differentiable manifold setting, it's never meant that.

Mike Miller

What is probably likely is that implicit in the author's mind is the case of a codimension 1 orientable thing.

Certainly that's the only case needed here.

Ted Shifrin

01:51

Yeah, don't over-generalize from this. I don't think it was used in general settings that way.

As far as I know, embedding always meant diffeo to its image :P

Mike Miller

This is about tame stuff, though. What's written here is "an embedding is said to be smooth if..."

That's what I was saying I was amused by.

Ted Shifrin

I think this is local terminology ...

Committing to a challenge

@ABeautifulMind We rock

Mike Miller

Local to the paper, or local to a point?

Ted Shifrin

If you look at Milnor's differential topology lectures from the late 50's or early 60's (which I have somewhere), I think he is consistent with modern use.

local to the paper, @Mike.

Mike Miller

01:52

fair 'nuff

Ted Shifrin

Yeah, Milnor's 1958 lectures say embedding = immersion that is a homeomorphism to its image.

Committing to a challenge

It will have been an hour, so nap time @Abe

Mike Miller

Thanks for checking, @Ted.

I have probably spent too much time goofing off in here. Back to the cave.

Ted Shifrin

You had me worried ... Bubye. Say hi to Pedro in the morning.

ᴇʏᴇs

Morning @Mike

A Beautiful Mind

01:55

@Committingtoachallenge I am thinking if I should go now or wait another hour.

Committing to a challenge

Cya @Mike

Mike Miller

I won't see Pedro for a continuum of mornings from now, @Ted, given it's always morning in LA

Committing to a challenge

@Mike À plus tard

Ted Shifrin

well, I won't tell you to say good night to Pedro :)

Committing to a challenge

@ABeautifulMind I will be gone in an hour for an hour

Ted Shifrin

01:56

A bientôt, @Committing.

A Beautiful Mind

@Committingtoachallenge That means now?

Committing to a challenge

@ABeautifulMind in an hour for the following hour

@TedShifrin Le close enough(sic[it's a meme])

A Beautiful Mind

@Committingtoachallenge I think I will wait another hour.

Committing to a challenge

@Ted Am I just trying to count the number of subsets of size $k$ of $\{1,\dots,N\}$ for the dim(W(k))?

Transcript for

$Mathematics$ Mathematics