Mathematics - 2017-05-11 (page 3 of 3)

Jason Bourne

22:00

Too many Erics in this chat, no idea who is who, lol.

Balarka Sen

there's an Eric, and there's an Eric Stucky

two many, not too many

Jason Bourne

You like puns don't you? Actually I don't, though I like language.

Balarka Sen

I like the good ones

Jason Bourne

I think that word puns are like math exercises to me.

Eric

@Jason a lot of them were comparison Theorems yeah

Jason Bourne

22:02

Every James Bond movie has lots of puns.

@Eric There is even a book called Comparison THeorems in Riemannian Geomtry by Cheeger and Ebin, lol.

Balarka Sen

i remember watching lots of bond movies (because family members with not that great taste) but i didn't like them. not enough to remember any of them at the very least

Eric

I am aware of it. Been told it's good

Jason Bourne

@BalarkaSen Not that great taste? LOL

Balarka Sen

@MikeMiller Heh, it seems like Solyaris (72) was marketed into US as "answer to Stanley Kubrick's 2001" (69?)

amuses me to no end

Jason Bourne

@BalarkaSen Have you watched Blue Velvet?

Balarka Sen

22:05

Not yet. But actually I might right now. It's not that late.

Jason Bourne

@AkivaWeinberger I wonder who still reads Euclid today. Is it even worth reading? I don't know.

@BalarkaSen I recommend Blue Car and Blue Desert as well.

Balarka Sen

i don't watch movies based on whether or not the title contains blue, unfortunately

Eric

@Jason I read Euclid when I was in high school. I would say it's worth it if you're a history nerd, but not super worth it for learning math.

Akiva Weinberger

@JasonBourne I read a bit at the start. Maybe I should pick it up; I'm horrible at geometry.

(Re: Eric's comment) (Or maybe not…)

Jason Bourne

@BalarkaSen Looks like everything I say sounds so stupid. =)

Eric

22:07

Yeah I think there are better places to learn that stuff

Akiva Weinberger

@JasonBourne Is English your native language? (Out of curiosity)

(I'm guessing yes?)

Jason Bourne

Besides Lee's Axiomatic Geometry and Hilbert's Foundations of Geometry, there is also Martin's Foundations of Geometry and the non-Euclidean plane. Martin wrote a lot of interesting geometry books.

@AkivaWeinberger Yes, I think it is.

Akiva Weinberger

Where're you from, America?

Jason Bourne

Nope, I live in Asia, in a not very good place.

Balarka Sen

@JasonBourne Nah, just that I'm choosy with what movies I watch

or what books I read, or ...

Jason Bourne

22:11

@BalarkaSen Certinaly not as choosy as me since I take years to think about books. =)

However, I live in one of the richest countries on earth, but money is not everything. In fact, money is the least important of all important things.

Akiva Weinberger

Singapore?

Jason Bourne

My location is a pseudo secret for pseudo secret reasons. =) I just tell people I live in Antarctica, as a joke and a cover.

Akiva Weinberger

It's either Singapore or Hong Kong.

Jason Bourne

I no longer consider where I live home. It's just a place I happened to be born in, and I hope to leave as soon as I can.

Balarka Sen

I like to say I live in one of the most nothing countries on earth, where we have the most amount of nothing, and the most nothingest peoples, and the most densely populated nothings per square kilometers. But I kinda like it.

That's also an irrelevant rant to a large extent though

Jason Bourne

22:19

Well, the best thing about India is it is the country where the Buddha lived and taught, though not where he was born.

There are only two people I really listen to. One is my mum, and the other is Buddha.

Balarka Sen

I don't know anything about Buddha's philosophy except the little there is on the Jataka epics.

Jason Bourne

The Jataka takes don't talk much about the teachings at all.

Balarka Sen

Yep.

Jason Bourne

I guess I have been hurt by so many people around me I no longer trust anyone.

I guess that's my rant.

trilolil

Hello everyone, I am trying to understand something about Taylor series:

f(x, y, z) has the following taylor series: $\frac{\partial{f}}{\partial{x}} x+ \frac{\partial{f}}{\partial{y}} y + \frac{\partial{f}}{\partial{z}} z$

But according to this source (http://docs.opencv.org/trunk/d7/d8b/tutorial_py_lucas_kanade.html) the taylor series should have the following form:

$\frac{\partial{f}}{\partial{x}} x+ \frac{\partial{f}}{\partial{y}} y + \frac{\partial{f}}{\partial{z}} = 0$

where $x = \frac{dx}{dt} $ and $ y = \frac{dy}{dt}$

Could someone explain me why x and why equal derivatives with respect to time and aren't just constants?

Akiva Weinberger

22:26

Hah:

123

A: Does notation ever become "easier"?

As others have pointed out, it gets much better if that's your first semester. But in my experience, there is not much relief between, say, years 2 and 4 of your studies. Sure, you get more mature, but the material gets more difficult too. To address your question whether "more advanced mathema...

(Re: the Arnold quote)

Balarka Sen

so true man

trilolil

Anyone any idea?

Balarka Sen

textbook should include awkward lines explaining what passages of symbols mean

even though it's awkward

Akiva Weinberger

@trilolil No idea

trilolil

@AkivaWeinberger thanks for your reply

anyone else?

Akiva Weinberger

22:30

@BalarkaSen Sometimes I feel like I want to say it the way I'd say it out loud. But often, I can't quite say it out loud (where, in a face-to-face conversation, I'd start looking for something to write with)

Balarka Sen

Huh. My hands usually fly around in every direction when I talk math face-to-face, that also helps

can you give an example?

trilolil

@AkivaWeinberger I already understood it. So stupid

it s not a derivative but simply a division...

Akiva Weinberger

@BalarkaSen An example where I would be able to say something out loud, differently than my first instinct with writing? None comes to mind, actually…

For an example that I wouldn't be able to say out loud: an analysis proof with lots of integrals I guess

Balarka Sen

Oh, that's strange.

Akiva Weinberger

I dunno, maybe we should do some math and if an example comes up I'll point it out :P

If I read a textbook like Rudin out loud, I often paraphrase stuff.

Balarka Sen

22:34

@AkivaWeinberger What's the basic idea behind excision theorem?

Can you say this out loud (if you remember) without writing anything whatsoever?

Akiva Weinberger

Oh, god…

I don't quite remember, no

Balarka Sen

I am asking 'cuz I can. This is actually very interesting; I wonder if you're fundamentally an algebraic person more than a geometric.

@AkivaWeinberger Ah, ok. Forget that then.

Akiva Weinberger

@BalarkaSen Geometric, probably

> I am asking 'cuz I can.

Can remind me how it goes, then?

Balarka Sen

That should mean your first instinct is to wave a lot of hands and try to describe the shape and size of the things than write the symbols!

Akiva Weinberger

Maybe I don't have enough face-to-face conversations about math.

They all tend to be on here.

Balarka Sen

22:39

@AkivaWeinberger OK. Do you want me to do it without symbols or with? :) The latter would probably take less time over chat :P

Akiva Weinberger

With, sure

Balarka Sen

Say $X$ is your space with an open cover $O$. $C_*(X)$ the singular chain complex and $C_*^O(X)$ the "excision" chain complex, consisting of singular simplices the images of which fit inside some open set in $O$. Remember that excision is equivalent to saying the inclusion map $C_*^O(X) \to C_*(X)$ is a chain homotopy equivalence (or in particular, is an isomorphism in homology)

It's a 2 line argument to see why this implies the standard formulation of excision

In any case, the idea is simple. Construct a map $C_*(X) \to C_*^O(X)$ (candidate for the "chain homotopy inverse").

This is given by taking a simplex, subdividing it like mad till you exceed the Lebesgue number of the cover $O$ for each subdivided piece, and you get a chain in $C_*^O(X)$. Extend linearly.

That's all.

Akiva Weinberger

(Went for a minute; I'm back)

@BalarkaSen Ohh. Right. Duh.

Ignore that

Balarka Sen

Well, $H_*(X, A) = H_*(X/A)$ for a good pair $(X, A)$.

That's a consequence anyhow, not the formulation

Akiva Weinberger

Excision was $H(X-Z,A-Z)=H(X,A)$.

Balarka Sen

22:49

Yup, Z subset A subset X, closure of Z contained in interior of A

I don't remember the 2 line argument :P Probably you look at the open cover {int(A), X - cl(Z)}

Yeah that's exactly it

Akiva Weinberger

OK. Yeah, I could've totally said that out loud if I actually remembered what the excision theorem was :P

(I remembered the theorem but forgot it was called the excision theorem)

…and if I remembered the two line argument.

Balarka Sen

Ahh.

Well you have surprisingly good geometric intuition even if you can't say stuff out loud so it doesn't matter much

Akiva Weinberger

Ah, thank you :)

Balarka Sen

weren't we supposed to learn differential forms at some point together? i'm terribad at it

Daminark

Hey

Akiva Weinberger

22:55

Oh. Right. Yeah, I haven't looked at Ted's book in a while (don't tell him!)

Balarka Sen

@Daminark Speak of the devil. I mean, demon.

Akiva Weinberger

But I haven't actually gotten up to forms yet

Daminark

Ayyy

Balarka Sen

@Akiva Are you through inverse function theorem stuff?

Akiva Weinberger

@BalarkaSen Dammit, I was just about to say that edit

Balarka Sen

22:56

:P

Daminark

Ah, we're talking about forms, I see

Fun stuff

Akiva Weinberger

@BalarkaSen Yeah, I'm on section 6.3 "Manifolds Revisited"

Balarka Sen

Not demonic forms

only differential forms

Daminark

I was thinking of Platonic forms

Balarka Sen

@Akiva aha

psst. I didn't really do 7 carefully

Daminark

23:00

Multi integration?

Balarka Sen

there are some cool exercises in there however; I didn't read the proof of change of variables although I understand the idea

@Daminark Yep

Daminark

Lol I'm doing most of those theorems in a Lebesgue integration context and I'll probably just call it a day

Because the Lebesgue stuff seems nicer to deal with than Riemann

Balarka Sen

needle point life blinds the will to be next / call it a day, call it a day

Akiva Weinberger

Huh, I've never worked with Lebesgue (though I know what it is)

Balarka Sen

I don't know anything about measures to be honest. Beyond what they are.

I was told in the last week of my informal reading course on Fourier theory about Borel, Lebesgue, how to integrate with them, etc etc but I forgot it all

Daminark

23:05

So basically, the idea is that you define a simple function as being a linear combination of indicator functions

(of measurable sets)

So $f(x) = \sum_{i=1}^n c_i\chi_{A_i}(x)$

You define the integral of that to be $\int f = \sum_{i=1}^n c_i\mu(A_i)$

Then if $f$ is non-negative, you define $\int f = sup(\int g)$ where $g$ is a simple function and $g \le f$

Lozansky

Noob question

Any easy way to get the eigenvectors of this

\begin{bmatrix} 3x_1^2-r^2 & 3x_1x_2 & 3x_1x_3 \\ 3x_1x_2 & 3x_2-r^2 & 3x_2x_3 \\ 3x_1x_3 & 3x_2x_3 & 3x_3^2-r^2\end{bmatrix}

Daminark

If you're not necessarily non-negative, you split up $f$ into its positive and negative parts, $f^+$ and $f^-$, and then $\int f = \int f^+ - \int f^-$

s.harp

have you tried asking mathematica

Lozansky

Well, by hand

Daminark

$f$ is integrable if at least one of those values is finite

Balarka Sen

23:11

damn I was fiddling with my pen and it flew 5 feet across the bed

s.harp

well first rescale all the $x_i$ by $\sqrt3$ to get rid of those annyoing factors

Balarka Sen

don't feel like fetching it

Daminark

Lel @Balarka

s.harp

next see that you are subtracting $M_{ij}=x_i x_j$ by something proportional to identity

so you only need to find the eigenvectors of $M_{ij}$

$a_i=\lambda x_i\sum_jx_j a_j$ is the equation for the coefficients of the EV $a$

Lozansky

@s.harp Not sure I follow

Adeek

23:17

hi everyone

Daminark

Hey @Adeek!

Adeek

Does anyone know how if I make some table

Lozansky

$\det(A+B) \neq \det(A)+\det(B)$

Adeek

and I want to connect two things from a table that is draw a line from A to B

in a table

is it possible ?

Daminark

What's the definition of table?

Akiva Weinberger

23:24

Are you talking about LaTeX?

@Daminark (/r/nocontext)

Daminark

Lol right?

Lozansky

@s.harp I understand that $M_{ij}= x_ix_j-\delta_{ij}r^2 = x_ix_j-r^2 \mathbf{I}_3$ but I'm not sure how the eigenspace of $\mathcal{M}_{ij} = x_ix_j$ is the same as the eigenspace of $M_{ij}$. Or is that not what you are claiming?

Akiva Weinberger

They're related, but they're not the same.

Lozansky

It's very late here so I might just as well be blind

Right

Akiva Weinberger

@Lozansky You want to see where $\det(M-\lambda I)=0$, right?

Lozansky

23:27

Yeah

Akiva Weinberger

So that's the determinant of $x_ix_j-(r^2+\lambda){\bf I}_3$

Lozansky

Yes

Akiva Weinberger

Now divide through by $r^2+\lambda$ and substitute $x_i=x'_i\sqrt{r^2+\lambda}$.

You end up with $(r^2+\lambda)^{1/3}\det(x'_ix'_j-{\bf I}_3)$, I think.

So focus on the eigenvalues/vectors of $x'_ix'_j$.

I'm not sure how you find that, but it's certainly much simpler than what you started with, right?

Oh, wait, hold on.

Now I'm confused. $\lambda$ is outside the determinant? I probably messed that up…

…unless $\lambda=-r^2$ is the answer…

Lozansky

Its not

Wait

Akiva Weinberger

Yeah, the $x'$ depend on $\lambda$ in a nontrivial way, so it's gonna be more complicated.

Lozansky

23:32

My original matrix is $M_{ij} = r^{-5}(3x_ix_j-r^2\delta_{ij})$

That what I wanna find the eigenvalues/vectors to

Akiva Weinberger

So ignore all that. (And $\lambda=-r^2$ makes no sense because in that case the definition of $x'$ involves a division by $0$)

6 mins ago, by Akiva Weinberger

So that's the determinant of $x_ix_j-(r^2+\lambda){\bf I}_3$

^That part should still work

Lozansky

Yeah that's clear

Akiva Weinberger

So they're $r^2$ less than the eigenvalues of $x_ix_j$

Lozansky

Uhmm

Akiva Weinberger

No?

Lozansky

23:36

Linear algebra was a long time ago. That might actually make sense

But I mean $\det(A+B) \neq \det(A)+\det(B)$ generally

Isn't that what you are doing?

Balarka Sen

"zen buddhism and the art of motorcycle maintenance"

man this must be fun to read

Eric Stucky

23:54

^ this is a very good book ^

Transcript for

$Mathematics$ Mathematics