Mathematics

9:00 PM

Sorry, I mis-typed it:

I(X, A) = \sum_{\ell} O\left( \Pr[A=\ell] \left(1-\frac{\Pr[A=\ell\mid X=0]}{\Pr[A=\ell\mid X=1]}\right)^2 \right)

Ward Beullens

I am working on a proof, im almost there, but i need a little help in the final step

Evinda

@ClementC. I wanted to ask you if you have an idea about this: math.stackexchange.com/questions/1759898/distance-of-bch-code

Mambo

Which proof

Evinda

Or weren't you referring to me? :) @ClementC.

Clement C.

@Evinda Sorry, cannot really help you there :/

Evinda

9:01 PM

A ok...

Mambo

@WardBeullens what proof are you studying?

Ward Beullens

@Mambo The last thing i need is to finish a proof is the following:
Given a homotopy T : [0,1] X M -> TM starting from the zero section, there exists an epsilon such that for all t in [0,epsilon] T(t,\cdot) is the graph of a vector field.

I hope I am making myself clear.

Mambo

Can you give a link to the proof you are referring ?

Clement C.

@Mambo if I understood correctly, it's a proof he is writing, not one he is reading

Ward Beullens

Indeed, I am making a proof myself for a homework.

Clement C.

9:09 PM

(sorry to ask again, but no one has a clue about how I should start to prove this inequality? I guess there is some trick i'm not seeing, to get rid of the logs and make this appear...)

I could go on taking this for granted, but that's a crucial part of the proof, and it seems wrong to just build on it without understanding where it comes form.

Ward Beullens

@ClementC. I cant help you, sorry

Mambo

@WardBeullens What about $M$ and $TM$?

Ward Beullens

M is a smooth manifold, and TM is it's tangent bundle

You can assume that M is compact if that is necessary

It probably is necessary

Mambo

$TM $ is the set of all tangent vector fields, right?

Ward Beullens

Its the Set of all the tangent vectors

Mambo

9:16 PM

Oh sorry, yes

Anthony

Ayo

the gradient of a vector wrt to a vector is a matrix?

How do you use the matrix to update the vector?

anon

@WardBeullens presumably you want to say T(t,-) is a section of TM->M for all t

@Anthony huh?

Balarka Sen

@Anthony what does gradient of a vector wrt a vector mean?

Anthony

Yeah I just said a lot of nonsense.

anon

it means $\partial f_i/\partial x_j$ @Balarka

by update do you mean provide a linear approximation for $f$ in a nbhd of the original $x$?

Anthony

9:22 PM

Yeah

I guess I have to choose a direction?

Balarka Sen

That's called gradient of a function, last I heard.

anon

that would be $f(x+h)\approx f(x)+f'(x)h$, where $x,h,f(x)$ are vectors and $f'(x)$ is the matrix

Anthony

I guess that makes perfect sense. What is up with my head lol

anon

@BalarkaSen well, yes, a vector function is indeed a function, but we can distinguish from the gradient of a scalar function

Mike Miller

morning @anon

anon

9:23 PM

morning

Balarka Sen

edit out "gradient" by "jacobian" in my comment.

Ward Beullens

@anon no, i dont need that it suffices fot t in [0, epsilon] In general it's not true i think for all t

anon

@WardBeullens Consider [0,1]xS^1 -> TS^1 defined by (t,x)->(R(t)x,0), where R(t) is rotation by t radians. This is a homotopy of the zero section S^1->TS^1 but no T(t,-) is a vector field for any t>0 no?

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