Speaking of which here is your random topology fact of the day! If $U$ is a nonprincipal ultrafilter on $\Bbb N$, then $U$, as a subset of the product space $2^{\Bbb N}$ cannot be Borel. In fact it doesn't even have the Baire property
The set of continuous functions $C[0, 1]$ is obviously not measurable in the product sigma-algebra $([0, 1]^{\Bbb R}, \mathcal{B}_{[0, 1]}^{\otimes \Bbb R})$. Is it measurable in the Borel sigma-algebra of the product topology of $[0, 1]^{\Bbb R}$?
Someone told me that the Riemann Hypothesis is equivalent to
$$\prod (1 - 2/v_i) = \frac{\pi}{6}$$
where the product is over all the nontrivial zero's $v_i$.
( for the product we take conjugate pairs of nontrivial zero's ordered by size )
Is that true ?
If true, how to prove it ?
Can this idea be...
I was like "Fubini's theorem for categories, that sounds awesome", but then I realized it's category as in "of the first/second category" and not the cool categories :/
if your manifold lives embedded in R^n and you identify the tangent space with the appropriate affine subspace of R^n, you can actually get a chart of your manifold by orthogonally projecting to the tangent space
I don't think this is useful, but it's nice to know
@AlessandroCodenotti But "being oscillation zero at every rational" is not an event that literally depends on countably many points on $[0, 1]$, right?
@Balarka I was told that a smooth structure on R^4 is diffeomorphic to the standard one iff it turns addition into a smooth map, but I can't find this with a primitive google search. Does this have a name/do you know a reference?
I didn't know that, surprising. I guess if both addition and multiplication by $-1$ was smooth under the smooth structure on $\Bbb R^4$ it would become a Lie group and therefore not exotic
Lie groups have very rigid smooth structures; they are automatically canonically analytic for example
We have discussed something similar before. Balarka, do you remember the example of two bundles that are diffeomorphic as manifolds but not isomorphic as bundles?
I meant the tangent bundle doesn't recognize smooth structure in the sense it can go either way. Exotic spheres always have isomorphic tangent bundles, for example, I believe
@Thorgott We should actually read some microbundle theory
What's an example of two non-isomorphic vector bundles $E,F$ over the same base such that the total spaces $Total(E), Total(F)$ are homeomorphic? Assume that rank of these bundles is the same as dimension of the base manifold.
(EDIT: Mike Miller brought up the excellent point that I need to loo...
@Thorgott Somehow the discussion is centering on what you observed earlier. Here's the deal; let $M$ be a topological manifold. $T_p M$ makes no sense, but why can't you just declare $T_p M$ to be a small chart around $p$? Choose an atlas $(U_i, \varphi_i)$ on $M$ and try to construct an uhhhhhhh "bundle" over $M$ with fibers being these charts $U_i$ (not a vector bundle because no canonical vector space structure)
How do you do this? Well, over $U_i$ it should be the trivial vector bundle $U_i \times U_i$
So patch them togather, do $\coprod U_i \times U_i/\sim$ where $(x, v) \sim (x, \varphi_{ij}(v))$, where $\varphi_{ij}$ are the literal transition functions
And thusly you have a bundle $E \to M$, projection being projection of $U_i \times U_i$ to first coordinate, with $\Bbb R^n$ fibers. NOT a vector bundle, and depends on the atlas
This is called the tangent microbundle; an analogue of the tangent bundle for topological manifolds
Here the transition functions are just manifold transition functions, given by the atlas
With some more thought you can convince yourself $E$ is a small neighborhood of the diagonal copy of $M$ in $M \times M$. In the smooth world, the normal bundle is indeed isomorphic to $TM$
So that's an alternative, tricky way to phrase what the tangent microbundle is. Given a top manifold $M$, take a nbhd of $M$ in $M \times M$ and declare that to be tangent bundle
microbundle equivalence is exactly like bundle equivalence but germinal around the zero section; you don't care about global things
because the problem of whether a topological manifold admits a smooth structure can be understood as an algebraic problem of when the tangent microbundle comes from an actual tangent bundle
microbundles were developed by Milnor, and was used by Kirby and Siebenmann to come up with their invariant
and that relates back to the kind of questions you were asking; how are tangent bundles of different smooth structures related?
you have an underlying topological tangent bundle so one would expect that to say something about that
@BalarkaSen This is sort of nonsensical as written because $\varphi_{ij}$ is partially defined on $U_i$. Anyway I know how to fix this but I won't write it unless anyone's ACTUALLY curious
Er I mean so $TU_i = U_i \times U_i$, right? The first copy is the base of the tangent bundle and the second copy is the fiber copy, should be identified with $\Bbb R^n$ by $\varphi_i : U_i \to \Bbb R^n$ chart homeomorphism. The idea is to build $TM$ from $TU_i$'s by patching these guys along the base copies and identifying fibers completely
@TedShifrin I understand the inequality thanks. If you start at the neighbourhood of the so called infliction point then your solution will tend to the infliction point or else it is unstable because it will diverge.
So if $x \in U_i \cap U_j$, I want to glue the fiber copy of $U_i \cong \Bbb R^n$ hanging out above $x$ in $TU_i$ with the fiber copy of $U_j \cong \Bbb R^n$ hanging out above $x$ in $TU_j$
Hello! @Sebastiano asked a question regarding, kind of, non-standard integrals (meaning highschool students rarely encounter those examples). The OP, as far as I'm concerned, would like to know if there are some examples he should memorize and how to recognize integrals which require some "advanced" methods. I thought of asking here because the question might get closed due to lack of clarity and context. In the meantime, the OP included some more information.
"Is there a criterion, a clue that makes me think that certain integrals can also be solved through complex integration and how to solve them?"
When I can't solve an integral for my students of an high school, I use the numerical methods.
If I have these integrals how are they solved used the com...