Mathematics - 2013-08-25 (page 1 of 2)

Peter Tamaroff

01:07

@TedShifrin

Howdy.

Kevin Driscoll

It seems I may have converted my integral equation into an integro-differential equation...... bad times

Peter Tamaroff

01:23

@KevinDriscoll Run. Don't look back.

Kevin Driscoll

Haha, that's my plan

I'm not positive yet though, need to do more calculating

Peter Tamaroff

01:50

@IanMateus Do you know about differential forms?

Ted Shifrin

03:15

@Peter: You rang?

Peter Tamaroff

@TedShifrin Yes!

=D

I have worked the properties of the exterior derivative already.

Ted Shifrin

How was tennis? I play at 8 am, so only here briefly ....

Ok ....

Peter Tamaroff

@TedShifrin Good. I have changed from low pressure to full pressure balls and my "student" (??) is doing awesome. I am very happy.

Plus, I get to hit the ball harder and faster, which makes things less boring.

Heh!

Ted Shifrin

Young kid?

Peter Tamaroff

@TedShifrin Nope, he's 50.

Ted Shifrin

03:17

Oh ... I've always used regulation balls :)

Peter Tamaroff

@TedShifrin What is that?

Ted Shifrin

Penn balls they use for matches ... No low pressure nonsense :)

Peter Tamaroff

@TedShifrin Oh, sure! But you must understand that low pressure balls are fantabolous for starters.

Specially in a fast surface.

Ted Shifrin

That must have been invented long after I startedv:D

Peter Tamaroff

@TedShifrin Yes, sure.

I think the French started the "Play and Stay" wave.

And the "new wave" of teaching, the one I learned.

Ted Shifrin

03:20

I like faster ... Then I can hit volley winners sometimes :)

Peter Tamaroff

@TedShifrin Heh, sure. Today I got to do some awesome shots and serves. Sent a cone flying =)

Ted Shifrin

Ok, back to your question.

Peter Tamaroff

@TedShifrin I am facing the "Poincaré Lemma".

Ted Shifrin

Yes ...

Peter Tamaroff

@TedShifrin Well, the proof is some humongous computation. What I am trying is to see how things work in small cases so to understand what is going on.

So I am trying to work out the case $\omega$ is a $1$-form $$=\sum_{i=1}^n\omega_i dx_i$$

Ted Shifrin

03:22

The case for 1-forms is done explicitly in my book.

Peter Tamaroff

If I am not doing things incorrectly $$d\omega=\sum_{i<j} (D_i\omega_j-D_j\omega_i)dx_i\wedge dx_j$$

@TedShifrin Oh! Let me look.

Ted Shifrin

Better to write $a_i$ for the coefficients so you don't get confused.

Oh wait ... By Poincaré lemma I thought you meant the converse. You mean $d^2=0$?

What fo you mean?

Do

Peter Tamaroff

@TedShifrin I mean that any closed form on a star shaped open set is exact.

Ted Shifrin

Ok, good. We agree. And, yes, I do exactly that in section 8.3.

Peter Tamaroff

Apparently this has some higher analog when we have a "contractible manifold".

@TedShifrin OK. I was having a hard time finding it.

Do you know you can give "hyperlinked" indexes to pdfs?

Ted Shifrin

03:26

Yes ... And I have the general case as an exercise in section 8.7, Inthink

Peter Tamaroff

@TedShifrin OK.

Ted Shifrin

Yes, but this is only a .pdf for you. The LaTeX file was used by the publisher to make the book.

Peter Tamaroff

@TedShifrin Ah, OK. =) Sorry.

(Now I feel bad)

Ted Shifrin

After I sent it, I discovered a bad page break :)

Have you found it?

The page in 8.3, I mean ....

Peter Tamaroff

@TedShifrin Can I show you my computations?

Ted Shifrin

03:31

Proposition 3.3 in chap 8 ... It'll be torture to type and read at this hour. If you email it to me, I'll get back to you after tennis :)

Peter Tamaroff

@TedShifrin Wait, how is it near 8 a.m. there?

Ted Shifrin

No, no, it's bedtime :)

Peter Tamaroff

OK. I'll TeX 'em up.

@TedShifrin Ah. OK. It is 12:30 here.

Ted Shifrin

You're still young ... You don't need sleep :) I'm old and have had cancer and heart surgeries ... SomI do :D

I'll get back to you in the morning ...

Peter Tamaroff

@TedShifrin Sleep tight, Ted. I was just telling you what timezone I was in. =)

Ted Shifrin

03:34

Yeah, you're east of me, but I never visualize it that way.

Night night :)

Kevin Driscoll

Really? 12:30 in Argentina? I thought it wouldve been the same as eastern time

guess you're more west of me than I imagined

Peter Tamaroff

@KevinDriscoll I'm in Buenos Aires which is basically coastline.

skullpatrol

;-)

Peter Tamaroff

@skullpatrol Thank you. The GNs had their snipers ready to fire.

skullpatrol

:D

Kevin Driscoll

03:39

I had an arugment about grammar the other day

skullpatrol

Who won?

Kevin Driscoll

It was kinda stupid

We both lost

I lost respect for them because I think its silly ot have a prejudice toward "correct English" when plenty of intelligent people grow up in the South or black communities speaking non-standard english

it doesn't affec the quality of their ideas, so its just a bad prejudice imo

skullpatrol

Agreed.

Kevin Driscoll

He lost respect for me because I cant see the value in putting effort into how you say things and conforming to the established standard to make sure htat you are understood and as clear as possible

skullpatrol

hmmm...

...yep, you both lost respect.

Antonio Vargas

03:43

Hi all.

Kevin Driscoll

I understand though. I'm sure its statistically true that the people who regularly speak non-standard English dialects have a lower IQ than people who usually use standard english. I just don't think the difference is that large. And the actual ideas someone uses will tell you ll you need otr know about their intelligence. No need to treat their dialect as a proxy

Antonio Vargas

I remember seeing some really nicely handwritten course notes a while back. The prof used different colors and drew nice pictures. I'm pretty sure they're known for being so nicely handwritten. Does anyone know who they're by?

Peter Tamaroff

@AntonioVargas No clue, braw.

Kevin Driscoll

No sorry I don't

@PeterTamaroff You should be ashamed of that language :-P

Peter Tamaroff

@KevinDriscoll Haters gonna hate.

Kevin Driscoll

03:48

@PeterTamaroff Love it!

Personally, I prefer 'breh'

to 'braw' or 'brah'

Peter Tamaroff

@KevinDriscoll I approve.

Kevin Driscoll

04:00

UH OH. I do not approve of 2 delta funcitons multiplied together

Lucking I think they can be integrated individually because htey dont share all the same variables

Luckily*

Peter Tamaroff

@KevinDriscoll You can edit your messages.

Kevin Driscoll

AH yes I keep forgetting, thanks

Crypto

04:23

hello world

Crypto

04:41

does anyone want to talk real analysis?

Peter Tamaroff

@Crypto Maybe?

Crypto

I'm an undergrad and I'm taking a real analysis course right now and the lectures seem a but scattered, I just want to talk about some definitions

Peter Tamaroff

@Crypto OK.

Crypto

most recently we've been talking about lim sup

I have a general idea of what it means, the largest subsequential limit

is there anything else to it, excuse me if this is too broad

Peter Tamaroff

@Crypto What do you mean by "is there anything else to it"? What do you know about the $\limsup$?

Crypto

04:55

I know that it is the largest of subsequence of a given sequence, assuming that this given sequence is bounded

sorry largest limit of all subsequence

s

Peter Tamaroff

@Crypto OK.

And what else do you know?

Crypto

that's all

Peter Tamaroff

@Crypto Well, there are "more things" to the $\limsup$. You can seach MSE about it, and come back with questions.

Crypto

y do we study the lim sup of a set

Kevin Driscoll

Crypto, my understanding is that if the lim sup is finite

then you can say a lot of nice thing about the object

Peter Tamaroff

05:09

@Crypto Of a set? You mean $$\limsup\limits_{n\to\infty}A_n:=\bigcap_{n\geqslant 1}\bigcup_{k\geqslant n}A_k$$?

Success for Poincaré in the case $\ell =1$! @TedShifrin Hope you get happy when you read this.

Kevin Driscoll

05:53

@PeterTamaroff I'm happy for you

Peter Tamaroff

@KevinDriscoll Here's the code. You can paste it in MSE I guess.

@KevinDriscoll Did you get it?

Kevin Driscoll

I do see it, ya. But I'm quite sure I wouldn't understand any of it

Peter Tamaroff

There are some details I leave out because I think Ted has Spivak's book.

But they only concern the definition of $I$.

@KevinDriscoll Well, it is "simply" a version of $$\int_0^1 f'(tx)dt=f(x)$$ assuming $f(0)=0$ on steriods.

As Spivak says, "A somewhat involved calculation shows..."

That is really what it is. The wit is in the definition of $I\omega$.

Kevin Driscoll

Just takes a while to 'turn the crank'?

Peter Tamaroff

@KevinDriscoll Aha.

But, note how the case $\ell =1$ is slightly involved. I will look at $\ell =2$ tomorrow, for the knack of it.

Kevin Driscoll

06:06

Haha, yes. I hate it when the simplest case is complicated

Leaves little hope for the harder ones

Peter Tamaroff

@KevinDriscoll Heh, well, I think it is a matter of getting used to that.

Remember when the binomial theorem looked like some weird thing? If you're OK with it try a proof by induction on the multinomial theorem! =)

Kevin Driscoll

Good point

I think i'll file that under "things I probably can't do"

I'm quite bad at mathematical induction

I have very little experience with it

Peter Tamaroff

@KevinDriscoll "I'm quite bad at mathematical induction" + "I have very little experience with it" = My point exactly.

Kevin Driscoll

Haha, true

Peter Tamaroff

@KevinDriscoll OK, this is pretty fantabolous

Kevin Driscoll

06:14

I. Don't. Understand. @PeterTamaroff

Peter Tamaroff

@KevinDriscoll What is there to understand?

Kevin Driscoll

Uuuumm, exactly?

Peter Tamaroff

@KevinDriscoll What are you asking now?

I have to go. It is late.

I must chill meh brainz.

Bye byes.

N3buchadnezzar

07:04

yo

dafuq math.stackexchange.com/questions/468664/… $\mathrm{d}\pi$? :p

Jayesh Badwaik

@N3buchadnezzar What does $PV$ mean in sos440's solution?

Positive Value?

N3buchadnezzar

@JayeshBadwaik Principal Value

Principal value

In complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. Motivation Consider the complex logarithm function log z. It is defined as the complex number w such that :e^w = z\,\! Now, for example, say we wish to find log i. This means we want to solve :e^w = i\,\! for w. Clearly iπ/2 is a solution. But is it the only solution? Of course, there are other solutions, which is evidenced by considering the position of i in the complex plane and in particular its argument arg i. We can rotate count...

Jayesh Badwaik

07:26

Ohh, I see.

I did not know the standard notation.

N3buchadnezzar

=)

Dominic Michaelis

09:22

do you think i should add some more details here

anon

09:41

perhaps expand on what you mean by "use the binomial theorem"

are you talking about using the newton-binomial series? the power is fractional. or the geometric sum formula perhaps.

one can view it as $x(1+7/x+O(x^{-2}))^{1/5}-1)=x(1+\frac{7}{5}x^{-1}+O(x^{-2})-1)=7/5+O(x^{-1})$

Dominic Michaelis

i just use $a-b=\frac{a^2-b^2}{a+b}$

anon

how?

(blah^2/5 - blah^2/5) / (blah^1/5 + blah^1/5) does not seem very simplified

Dominic Michaelis

oh right we should use something different

anon

I would use $(1+c\epsilon+O(\epsilon^2))^\alpha=1+\alpha c\epsilon+O(\epsilon^2)$, which is newton-binomial or taylor series

Dominic Michaelis

yeah that is neater

anon

09:56

also the limit looks pretty much the same as x-> -inf

Dominic Michaelis

I did want to use $(x^4+x^3y+x^2y^2+xy^3+y^4)(x-y)=x^5-y^5$

anon

that might be doable

yes, indeed

you will simply move the limit down to a denominator and it will work out to 1+1+1+1+1

Dominic Michaelis

exactly

anon

but you want to call it the geometric sum formula rather than binomial theorem

Dominic Michaelis

it is nearly the same as binomial just more complicated :D

oh yes geometric sum formula might be better

newton binomial series is a bit to much for a calculus isn't it ? Although we proved it there :D

N3buchadnezzar

10:05

nah man, newton is cool

Dominic Michaelis

sure but it is cheating a bit isn't it =

skullpatrol

11:44

When our math teacher said, "We can go home but quietly... "

:D

Jossie

13:02

Math, not meth!

Gustavo Bandeira

13:16

Anyone aware of some question about sums of rationals and irrational numbers?

anon

most of them?

any question about sums whose terms are reals?

or perhaps you're talking about the fact that
rational+rational=rational
rational+irrational=irrational
irrational+irrational=either

Gustavo Bandeira

@anon Yes.

anon

then what more do you want to be said about it?

Gustavo Bandeira

But why is that? I have an intuitive answer in mind but I don't know if it's good enough.

The aperiodicity of a number summed with the periodicity of a number yields an aperiodic number, but I have no proof for that.

anon

the first is obvious, the second is obvious by contradiction, the third is obvious by cancellation/amplification

thinking in terms of decimal expansions can be a useful tool but it is hardly a universal go-to

Gustavo Bandeira

13:24

Yes.

anon

if rational+irrational=rational then irrational=rational-rational=rational, a contradiction

thus rational+irrational=irrational

also irrational+(itself)=2x(that same irrational)=irrational, whereas rational+(-itself)=0 is rational

Gustavo Bandeira

brb

anon

if you have familiarity with vector spaces, the rationals are a subspace of R and the irrationals are their complement

so ultimately this is a general fact about vector spaces (or even modules)

hhh

What is the name of this matrix operation?

$A=\begin{pmatix} 1 & 2 & 3 \\ 1 & 1 & 1 \end{pmatrix}, SomeOperation(A)=\begin{pmatrix}1*2*3 \\ 1*1*1\end{pmatrix}$, what is the operation SomeOperation?

Some sort of rowwise multiplication?

anon

hmm the latex isn't working for some reason

but anyway the answer is that that operation is too unnatural to have a name

hhh

13:34

Moved the question here so easier to see the question.

anon

you can use {\rm text} or \textrm{blah} to make non-italicized text in equations. also many prefer \cdot for multiplication over *, the star symbol is clunky and deprecated

user38268

13:54

@anon Man this shit about qcqs morphisms is confusing

user87637

15:36

@user38268 Why do you keep saying shit? LOL.

Ian Mateus

16:02

@PeterTamaroff probably not, unless you ask for elementary manipulations over some specific extensions of differentials (which I have been doing for a while). Why?

Einsteins Grandson

17:10

Anybody here?

If I need to find the line that is Intersection of two planes , how do I do it?

anon

cross product of planes' normals yields line's direction

skullpatrol

Hi @robjohn how are you?

robjohn

@skullpatrol I'm fine, how are you?

skullpatrol

@robjohn Fine, thanks :-)

Peter Tamaroff

18:21

@anon Dude.

$$d\left( {\sum\limits_{i = 1}^n {{\omega _i}d{x_i}} } \right) = \sum\limits_{i < j} {\left( {{D_i}{\omega _j} - {D_j}{\omega _i}} \right)d{x_i} \wedge d{x_j}} $$

Right or left?

@robjohn Too much exposition?

Ted Shifrin

@PeterTamaroff: You have mail!

Peter Tamaroff

@TedShifrin I have read it!

So I am going over it.

To see what is wrong.

Ted Shifrin

Okey dokey :)

I think you're off by a sign in the definition of $I(d\omega)$. I don't have Spivak's book here at home. I know how I usually do it, however.

Peter Tamaroff

@TedShifrin Oh.

Let me show you the definition.

Amber Roxanna

Hmm...is Spivak's book closer to a Real Analysis curriculum rather than an introductory calculus course ? Just curious. I've heard great things.

Peter Tamaroff

18:35

@AmberRoxanna Which of his books?

Ted Shifrin

Different book, Amber.

But, yes, Calculus is a bit tough for students who've never had calculus, and it is a true mathematics text, not a calculus text.

Peter Tamaroff

@TedShifrin Indeed. Awesome book.

Amber Roxanna

OK, and I was referring to his standard book titled "Calculus"

Peter Tamaroff

@AmberRoxanna If you can read it, it is very very good.

skullpatrol

For those capable...

Ted Shifrin

18:38

I have successfully taught a few students who've never had calculus the course using that book, but they were extremely talented. Generally, the students who take the course have had high school calculus first.

Amber Roxanna

I'm not smart enough for that book even though I learned calculus on my own.

skullpatrol

...for those not capable, it would be a waste of time, in my opinion.

Ted Shifrin

Hmm, @PeterTamaroff. Then your signs are all backwards on your second line, so the mistake has to come somewhere next.

@AmberRoxanna: Most students need a teacher teaching them that course. I find that a lot of folks on this site are trying to do self-study of things they do not have the background or maturity to do ... I've seen that a lot this summer.

Amber Roxanna

@TedShifrin Very true.

skullpatrol

Don't say "you're not smart enough"

Ted Shifrin

18:41

@PeterTamaroff: No, I was right. Check your signs from the definition.

@AmberRoxanna: But I would agree with @skullpatrol. You may very well be plenty smart and, if you're motivated and have the right guidance (teacher) you'd probably do great with it. But my experience with students in that course and in the multivariable analogue of it I wrote/teach all the time is that you must be really interested in understanding/doing proofs and in working hard.

Peter Tamaroff

@TedShifrin Hope I am not the case. =D

Amber Roxanna

@TedShifrin Well, at least most universities offer courses in introductory proof techniques.

skullpatrol

You need the background and/or maturity

Peter Tamaroff

@AmberRoxanna I am compelled to ask what's the need to include so many pictures of yourself in your profile. Specially when you're this young.

Amber Roxanna

@PeterTamaroff Oh, I really didn't have anything better to do.

skullpatrol

18:45

;-)

Ted Shifrin

Um, no, Peter, I'm NOT talking about you at all. You know tons of analysis, BTW, from what I've read from you. There are a gaggle of students trying to do Guillemin & Pollack's Differential Topology without knowing multivariable analysis (e.g., what you're doing in Spivak). And they've been trying to get us to do literally all their homework/test questions.

Amber Roxanna

@TedShifrin lol

Peter Tamaroff

@TedShifrin I am just teasing Ted!

Ted Shifrin

@PeterTamaroff: OK, I was right originally. Look carefully at his definition and what you wrote. You're off by a sign!

Peter Tamaroff

@TedShifrin OK!

Rewrites.

@AmberRoxanna So you're in highschool being taught that?

Or just reading on your own?

Ted Shifrin

18:46

OK, somehow I didn't think you were lacking in self-belief ... But we all make silly sign errors ... or not so silly ones. And in complex geometry, keeping track of $\sqrt{-1}$ versus $1/\sqrt{-1}$ is enough to drive one batty.

Amber Roxanna

@PeterTamaroff just reading on my own, mostly on the bus or train rides. When ever i get a bit of free time here and there

skullpatrol

@AmberRoxanna Welcome to the chatroom :D

Peter Tamaroff

@TedShifrin Heh, true.

@AmberRoxanna Ah, that's great.

Well, I hope you find my answer to your question useful.

Ted Shifrin

@AmberRoxanna: I'm teaching a very talented high school senior a one-on-one Spivak course this very year. We've made it through some of the hardest theoretical material just fine so far. Now we're getting to derivatives.

robjohn

@PeterTamaroff Looks fine to me...

Peter Tamaroff

18:48

@robjohn You can go ahead and delete my comment then.

Ted Shifrin

Do we all get to delete @PeterTamaroff? :D

Amber Roxanna

@PeterTamaroff thank you for answering

Peter Tamaroff

@TedShifrin I think only mods have Godly powers here.

Ted Shifrin

Oh, good :P I hope never to be one of those.

skullpatrol

@AmberRoxanna What grade are you in?

Amber Roxanna

18:51

@skullpatrol 10

Ted Shifrin

Wow ... and you've done calculus some already?

skullpatrol

*some calculus

;-)

Ted Shifrin

I meant it as an adverb, not adjective :P

Amber Roxanna

@TedShifrin yes, it's really not a big deal. I find it easier than pre-calculus actually.

Ted Shifrin

Manipulating formulas, or actually doing interesting application/word-problems?

robjohn

18:53

@PeterTamaroff I didn't see that there were three pictures there...

Amber Roxanna

@PeterTamaroff some idiot decided to delete my question. I should slap them now.

Peter Tamaroff

@AmberRoxanna No, the question is being closed as a duplicate. It is no big deal. Now I see you kinda asked this before.

skullpatrol

We don't "slap" here

Peter Tamaroff

@AmberRoxanna And careful: we know each other here! =)

Ted Shifrin

@skullpatrol: Speak for yourself :D

skullpatrol

18:54

Yes Sir...sorry

Peter Tamaroff

@TedShifrin Heh.

Ted Shifrin

@PeterTamaroff: You can tell I'm the typical boring math prof :P

Peter Tamaroff

@TedShifrin I had a teacher that threw chalks at us. I was hilarous. "You there shut up: BAM!"

Amber Roxanna

@TedShifrin I want to be a math professor but not a boring one.

skullpatrol

who says x and writes y

Ted Shifrin

18:56

Great, @AmberRoxanna. We certainly don't need more boring ones. @PeterTamaroff: I used to throw chalk, but I almost hit a guy in the eye, and immediately stopped doing it.

Now I throw puns, among other things.

2

Amber Roxanna

@skullpatrol I'm guilty of that

skullpatrol

:D

Ted Shifrin

@skullpatrol: What level are you?

skullpatrol

banana

Ted Shifrin

um ... that was a serious question :)

Amber Roxanna

18:58

@skullpatrol that's a level ?

Ted Shifrin

Probably in his video game mind, @AmberRoxanna :P

Amber Roxanna

@skullpatrol do you want a banana ? lol

Peter Tamaroff

@TedShifrin Ah, found the mistake. Bahoolas.

Ted Shifrin

You agree with what I said a hundred lines up? :)

skullpatrol

Where is Jasper when you need him?

Transcript for

$Mathematics$ Mathematics