Mathematics

9:01 AM

Well to take the example of the tangent bundle, the tangent bundle has the topology defined by... \begin{eqnarray}
\tilde{\psi}_U : \pi^{-1}(U) &\to& \mathbb R^n \times \mathbb R^n \\
(p, v^\mu \partial_\mu) &\mapsto& (\phi^\mu(p), v^\mu )
\end{eqnarray}

Balarka Sen

But a tangent bundle is a vector bundle, not a random fiber bundle

You started with fiber bundles

Slereah

true, but it is the tensor bundles I am most interested in

Balarka Sen

You'd have to reformulate the question for me then.

What are $E_1$ and $E_2$?

Slereah

Well eventually I'd like to show how to define a metric tensor on the glued manifold

So for a start I'm trying to show that the space of metrics from the original manifold is also one on the glued manifold

Balarka Sen

I do not understand what this has to do with the question you're asking me, but that's simple: If $M$ and $N$ are smooth manifolds with boundary $\partial M$ and $\partial N$ then $M \cup_{\partial} N$ also gets a structure of a smooth manifold. If you have Riemannian metrics on $\text{int} M$ and $\text{int} N$ you can use a partition of unity argument to get a Riemannian metric on $M \cup_{\partial} N$ too.

Slereah

9:07 AM

The trick is here that what I want is a metric such that $g|_M = g_M$

and same with $N$

ie the original sections must be preserved

Balarka Sen

Well my metric has that. It agrees with the metrics on the interiors of $M$ and $N$ outside a neighborhood of $\partial M = \partial N$

Slereah

alright, thanks

Balarka Sen

You cannot hope to preserve the metrics on all of the interior of $M$ and $N$

In general I mean

Slereah

Well there is a Trick to this

(if the boundaries are isometric, you can preserve them)

Alessandro Codenotti

Let $E_n$ be the number of elements of even order in $U_n=(\Bbb Z/n\Bbb Z)^\times$ (while $|U_n|=\varphi(n)$), is it true that $\frac{E_n}{\varphi(n)}\geq\frac12$ (in particular for $n=p^\alpha$, $p$ an odd prime) and that $\limsup\limits_{n\to\infty}\frac{E_n}{\varphi(n)}=1$? @Mathei

Balarka Sen

9:09 AM

@Slereah For sure, but you had given me no such restrictions.

Slereah

Also things are slightly tricker because the metric is Lorentzian :p

So sections aren't guaranteed

Hm, I think I might need specific conditions on the gluing, actually

The gluing should be on the base manifold and then extended somewhat to the fibers

not sure if that's unique, though

Dattier

@Alessandro : this problem 's your production ?

Alessandro Codenotti

Yes, but it's motivated by another problem I got as homework in algebraic number theory

Dattier

the level ?

licence or master ?

Alessandro Codenotti

Third (last) year of undergrad, but the problems are meant to be challenging, the one above would be a small step in the overall proof if it's correct

Dattier

9:23 AM

you don't know the answer ?

Secret

in The h Bar, 17 mins ago, by Akash. B

how did energy formed in space?

Is h bar invaded by some spiritual being?

I am seeing spiritual mumble jumble everywhere ever since he joined

Tobias Kildetoft

@AlessandroCodenotti The elements of odd order form a subgroup. What will the index of this be?

Dattier

@AlessandroCodenotti $\phi(n)=2^k \times E_n$ and $E_n \mod 2=1$

Alessandro Codenotti

@TobiasKildetoft The biggest power of $2$ which divides $\varphi(n)$ I think

Tobias Kildetoft

9:38 AM

right

Wow, 26 pages paper of which the final 6 are the bibliography

Alessandro Codenotti

Which means if $\varphi(n)=2^sm$ with $m$ odd we have that there are $m$ elements of odd order and $\varphi(n)-m$ elements of even order, and the last one is bigger or equal than $\varphi(n)/2$ because $\varphi(n)$ is always even (for odd $n$)

Tobias Kildetoft

right (you can rewrite the quotient you are looking at in a nice way using this)

Alessandro Codenotti

Indeed, thanks for your help!

Amorris

9:56 AM

How would you prove that a=4 and that x/=3?

Shobhit

Hello. On the wiki page of retract:en.wikipedia.org/wiki/Retract , its given that, a retraction is a continuous mapping from a topological space into a subspace which preserves the position of all points in that subspace, but the definition my professor wrote on the board included that the topological space is hausdroff. Is that compulsory?

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$Mathematics$ Mathematics