Mathematics - 2017-03-04 (page 1 of 3)

quallenjäger

00:00

Of a function at point x

Mike Miller

I've now abandoned the talk I was watching. Lasted until the last ten minutes.

Ted Shifrin

If you take $\omega = f(x,y,z)dx\wedge dy\wedge dz$ in $\Bbb R^3$ and three vectors $v_1,v_2,v_3$, then $\omega(v_1,v_2,v_3) = f(x,y,z)\times\text{signed volume of the parallelepiped spanned by } v_1,v_2,v_3$.

Semiclassical

hi chat.

Ted Shifrin

@quallenjäger: Not in general. Differential forms needn't come from differentiating functions.

Rehi @Semiclassic

Is this part of the conference, @MikeM?

Semiclassical

I watched a lot of practice 10-minute talks today, in prep for the annual March meeting of the APS.

Mike Miller

00:02

That's next week. Bob E gave a beautiful talk that I got lost at the end of.

Ted Shifrin

I think I gave 2 or 3 10-minute talks in my whole career. I hated talks that short.

Oh, OK, @MikeM.

I haven't seen him in a long while, either.

quallenjäger

@TedShifrin thanjs

Semiclassical

There was a neat one on certain states on a Penrose quasicrystal

quallenjäger

thanks!

Semiclassical

Probably some nice math there.

Ted Shifrin

00:03

@quallenjäger: You might want to watch some of my lectures on this stuff on YouTube if you're interested.

The link is in my profile.

quallenjäger

@TedShifrin how can I find it

Oh ok thanks

Ted Shifrin

There are 112 lectures altogether, but there are some explicitly on differential forms, integrating them, Stokes's Theorem, etc.

quallenjäger

Sounds interesting

Are you teaching at university?

Ted Shifrin

I retired recently. These were from classes a few years ago.

quallenjäger

Wow

Ted Shifrin

00:06

Do you think in terms of physics stuff at all, @quallenjäger?

quallenjäger

yes I am doing theoretical physics as minor

Ted Shifrin

Oh, so that'll be useful. I did a lot of physical interpretations in the lectures — work, flux, etc.

Maks

Hey, easy problem here
If I have P(x) and Q(x) polynomials
$P(x) = kx^3 + x^2 - k$
$Q(x) = x - 5$
I want to find a k such that P(x)/Q(x) has remainder 0
The equation should be $ \dfrac {P(x)}{Q(x)} = C(x) + 0$
Right ?

quallenjäger

Trying find equivalent of maxwell

Via differential form

Ted Shifrin

Oh, I didn't do that in the lectures, @quallenjäger, but there is a discussion of it in my book.

It's not hard.

@Maks, because $Q(x)$ is so simple, there's an easy way to do this.

If $P(x)=C(x)Q(x)$, what happens if you plug in $x=5$, @Maks?

Maks

00:09

Q(x) = 0

Ted Shifrin

So what does that tell you about $P(x)$ when $x=5$?

Semiclassical

Depends a bit if you mean the covariant formulation of it. Misner-Thornton-Wheeler's Gravitation is a standard physics reference for that.

Maks

0 ?

Ted Shifrin

Yup.

Maks

k can be every number

but what if x != 5 ?

Ted Shifrin

00:10

That's a cool book. There's also discussion in Abraham/Marsden/Ratiu, and other "math for physicists" type books.

Semiclassical

On the other hand when one is just doing vector electrodynamics then things are simple enough.

Ted Shifrin

I don't care, @Maks.

Frankel's book (The Geometry of Physics?) is really cool, too, @Semiclassic.

Maks

Why not ? Shouldnt I find an answer that works for every x ?

Ted Shifrin

@Maks: You skip ahead and get yourself lost.

quallenjäger

But vector for electrodynamics I will need rot and div, which depends on coordinate

I need something which is coordinate independent

Maks

00:12

@TedShifrin :O what ? ...

Ted Shifrin

But you can state all that stuff in terms of differential forms, invariantly, @quallenjäger.

That's in the lectures, too.

quallenjäger

That's why I'm looking for it.

Cool!

Ted Shifrin

You need to learn the math, but Frankel's book I mentioned above might be interesting to you.

@Maks: If you plug in $x=5$, you have to get $0$, so what must $k$ be?

quallenjäger

I will

Semiclassical

Arnold's book on classical mechanics is another reference, but that's a hard one.

quallenjäger

00:14

Maybe another thing you might know.

Ted Shifrin

It's beautiful, but it's typical Arnold.

quallenjäger

Is it possible to define stochastic differential equation as a bundle over a manifold?

Maks

Oh, I see it now, it does work for every $x$, I better shut up and get lost then hahaha

Daminark

Oh Arnold

Ted Shifrin

I don't know anything about stochastic differential equations, but differential equations can be defined in terms of sections of appropriate vector bundles, yes.

quallenjäger

00:15

how would I choose the manifold as the chain rule doesn't hold for Ito formula.

Daminark

When we were doing ODE I had started using it as a reference

That didn't last too long

Ted Shifrin

Demonark: You do know about Hirsch/Smale's beautiful book on dynamical systems and ODE?

Semiclassical

I think I may have seen that one.

quallenjäger

Lol Arnold isn't it a Russian book?

Daminark

Someone found a book by Perko after a while that ended up being aligned well enough with our treatment to be a good reference

Ted Shifrin

00:17

Yes, Arnold is Russian.

Semiclassical

Yep. But there are translations.

quallenjäger

Russian book are always elegant, but hard

Ted Shifrin

Check out Hirsch/Smale, @Daminark. It's fantastic.

Daminark

Hirsch-Smale is what they use in the ODE class here

I've heard good things about it

Though I discovered it after we had already moved on

Ted Shifrin

I wouldn't recommend it if it weren't worthy :D

Daminark

00:18

That's true

Ted Shifrin

Not that you trust me.

quallenjäger

@TedShifrin if the chain rule doesn't hold, is there a way to fix it by modifying the manifold?

Ted Shifrin

I don't really understand what that means, @quallenjäger.

Manifolds are just locally Euclidean space, and calculus is calculus.

There are structures, if you like, that are defined in differential geometry, that are not quite tensors, and so they do not transform according to the chain rule. But they're still geometrically meaningful. Connections (so-called Christoffel symbols you may have heard of) are an example of this.

quallenjäger

I mean the chain rule between manifold

$F:M \to N$ and $G: N \to O$ two smooth map

Ted Shifrin

You can't break the chain rule with manifolds. As I said, that's all local calculus, and calculus is calculus.

But I know nothing about stochastic stuff.

quallenjäger

00:28

@TedShifrin I am watching your videos, nice stuff!

Ted Shifrin

Thanks, @quallenjäger. Some will be slow for you — the class is mostly first- and second-year undergraduates (but very smart ones) in the US.

quallenjäger

ok I will need learn the math first I think:D

I find the pace quite accurate as it is very insightful

Ted Shifrin

Well, you know where to find me if you have questions.

quallenjäger

Sure

Daminark

I've always wondered about the whole European thing about specializing so early

I'm leaning that it's probably better here

On one hand, there's the fact that you can get so much farther in your own field, but somehow I think that there's a level of breadth that ought be present in education beyond that which is required in high school and such (required in the sense of, this is stuff that all citizens should know).

Also, it seems like this process starts in the last 2-3 years of high school, and in general asking people then what they want to go into, even loosely, is a bit much.

What do you think @Ted?

Zach Hauk

00:47

I'm back

Kasmir Khaan

reverse triangle inequality

Who has a cool geometric or algabraic intuition of thaT?

Zach Hauk

what's the reverse?

quallenjäger

Someone familiar with stochastic process here?

Daminark

Reverse triangle inequality says that $| \|x\| - \|y\| | \le \|x-y\|$

quallenjäger

@KasmirKhaan it just states that any site is greater than the difference between other 2 sites

*side

Zach Hauk

00:52

@Daminark hi

Daminark

How's it going?

Zach Hauk

im ok

umm, so i had a question

so space-time looks like this, right?

qph.ec.quoracdn.net/…

Would this be a manifold? Because at each point if you zoom in enough it looks like $\Bbb R^3$

quallenjäger

@KasmirKhaan this is quite obvious, as for a straight line this must fulfill

Daminark

Well, this is a representation of curved spacetime in 3 dimensions, but I think (and right now I'm really talking based on vague/speculative knowledge) it's an immersion of a 4-manifold in $\mathbb{R}^3$

Kasmir Khaan

@quallenjäger does it hold the same way in complex numbers?

Zach Hauk

00:55

I need to see a wrist doctor :(

Daminark

So spacetime with what's the Minkowski metric (basically, there's some sense in which time is "negative") is, I believe, a 4-manifold. Mass deforms it, but I think not enough that it stops being a manifold unless you're dealing with a black hole

Zach Hauk

I'm not a physicist so whatever :P

quallenjäger

@ZachHauk it is a curved 4d-manifold

Zach Hauk

that's what i thought

Daminark

Now, representing it in $\mathbb{R}^3$ is definitely not an embedding, I've heard of something called an immersion so I think it has to do with that

In terms of the manifold structure it's actually the exact same as the pre-relativity Galilean stuff, what matters is the metric

quallenjäger

00:58

@ZachHauk and it is far more complicated, as you cannot really move objects like in 3 dimensional space. These are collection of events

Daminark

Basically, the distance squared between two points is now $x^2 + y^2 + z^2 - t^2$

So now you get Lorentz transformations and a lot of fancy stuff

Zach Hauk

so it's kind of like how

Daminark

But now I should probably stop, I'm talking way out of my league with regard to differential geometry and physics. I think I haven't fed you any misinformation, but someone should definitely correct me if so.

Zach Hauk

if we took a surface in $\Bbb R^3$

and projected it onto the plane

Daminark

Pretty much

Zach Hauk

01:01

we'd have like more condensed grid where the surface is more bumpy

Daminark

01:12

So I started briefly down the wikipedia rabbit hole and it's cool stuff

LegionMammal978

Hmm

Was thinking of an alternate expression for standard deviation

If v_1, v_2, ..., v_n are the values:

stddev = sqrt((v_1^2 + v_2^2 + ... + v_n^2)/n - (v_1 + v_2 + ... + v_n)^2/n^2)

Simple

02:43

ocw.mit.edu/courses/mathematics/…

can someone explain me why $x_1,x_2$ are real solution on the first example

$x_1=e^{-t}\left([0;2]^T\cos(2t)-i[1;-2]^T\sin(2t)\right)$ and $x_2=e^{-t}\left([1;-2]^T\cos(2t)-i[0;2]^T\sin(2t)\right)$

satyatech

04:25

Hey guys ,I have asked a qn on main site. ,,,,.

Link

0

Q: Another interesting question on Multinomial;

I don't know how to proceed with this problem... But as this question was given in chapter of binomial so maybe Binomial theorem comes into usuage.

binomial-theorem

BAYMAX

This may be silly but try putting x = 0and see ?

many terms get 0

Semiclassical

That's not really going to help get the desired sum...

BAYMAX

ok...

Semiclassical

My hint: Given a pair of generating functions $A(x)=\sum_{n=0}^\infty a_n x^n$, $B(x)=\sum_{n=0}^\infty b_n x^n$, consider what the coefficients of their product $A(x)B(x)$ would be (multiply the first few terms out explicitly if it's not obvious). @satyatech

Actually, scratch that. The construction they're asking for is a bit different.

(A variation on the above idea should work, but it's not immediately obvious to me how it's done.)

satyatech

05:05

Yes the way you told me,I can't use that to implement here..

@Semiclassical

BAYMAX

wow a big question i got screwed me out of my mind!!!!!!

0celo7

What?

BAYMAX

A brain buster --------------- A monkey is at topmost of ladder ... (that is at first step) .and his aim is to reach at ninth step which is the water level...he performs this- he jumps forward three steps and then jumps backward 2 steps .. in how many jumps he would reach the water level ???....

any 1 ???

i got around 30 steps ??

i am very much excited about this... any1 got answer @0celo7

@DHMO

0celo7

05:22

I wasn't asking what the question is, I just think your wording was messed up.

BAYMAX

ok

is there any confusion regarding the question ??

ok guys its not a brain buster its brain enhancer .. please help

0celo7

Hmm. Seems I can't ping Huy.

BAYMAX

meaning ?

0celo7

He hasn't been in this room for 3+ days.

(something like that)

BAYMAX

k

0celo7

05:26

Well, I'm going to sleep. Good luck @BAYMAX

BAYMAX

good luck @0celo7 !

bye!

A monkey is at topmost of ladder ... (that is at first step) .and his aim is to reach at ninth step which is the water level...he performs this- he jumps forward three steps and then jumps backward 2 steps .. in how many jumps he would reach the water level ???....

Vrouvrou

05:41

hello

@AlessandroCodenotti are you there ?

@DHMO hello

DHMO

yes?

Compulsive Mathurbator

All he has to do is jump once. Then he falls down the ladder to water level. With any luck, he survives.

DHMO

@BAYMAX 1 -> 4 -> 2 -> 5 -> 3 -> 6 -> 4 -> 7 -> 5 -> 8 -> 6 -> 9 (11 jumps)

Compulsive Mathurbator

@BAYMAX Gravity is still a thing right?

Vrouvrou

@DHMO $f_A:E\rightarrow \mathbb{R}$ définie par

$f_A=\begin{cases} 1, ~~, x\in A\\ 0,~~,x\notin A\end{cases}$
$A\subset E$ est ouvert et fermé , et je doi montrer que $f_A$ est continue

est ce que c'est juste de dire:

Si $x\in A$ comme $A$ est ouvert lors $A$ est un voisinage de $x$ , et $f_A(x)=1$, donc $\forall \varepsilon>0, \exists V=A\in\mathcal{V}_x, f_A(x)=1\in ]1-\varepsilon,1+\varepsilon[$

d’où l continuité de $f_A$ en chaque $x\in A$ et je refais la mème chose pour $x\notin A$

@DHMO

DHMO

05:51

@Vrouvrou oui

Vrouvrou

vous pensez que c'est juste?

DHMO

oui

Little Rookie

What is the difference between Least square error and absolute error?

Vrouvrou

merci @DHMO

DHMO

@LittleRookie well one uses the square and one uses the absolute value

Little Rookie

05:53

Yea haha, i mean why is least square error more emphasized on?

DHMO

@Vrouvrou alors, pourquoi $A$ doit-il etre ferme?

@LittleRookie because it's easier to manipulate?

Little Rookie

when fitting data, we minimize the least square error than minimizing the absolute erorr

Vrouvrou

@DHMO pour que le complémentaire soit ouvert

et conc voisinage

DHMO

@Vrouvrou d'accord

Little Rookie

i see, absolute value are hard to work with since they are not differentiable at points.

DHMO

05:57

yes

Little Rookie

Oh yea, if i square the error formula, i can always take the square root of it in the end

Little Rookie

06:17

Supposed i want to find model a set of data with a non-linear autonomous system of ODE, how should i proceed to find the best model?
I could invent an model and fit parameters to it, then modify the model, fit parameters and compare the 2 models.
And repeat the above step many times simply by trial and error.

BAYMAX

07:10

nice @DHMO

What a comedy @CompulsiveMathurbator

Balarka Sen