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Dec
12
comment Prove that dim$($Ker$(A^n))$ is odd for all $n$ sufficiently large
could you better explain the entries of $A$.
Dec
11
revised Do the singular matrices form a topological manifold
edited tags
Dec
11
comment Compact space - definition
it has to be the case that every open cover has a finite sub-cover, not just that there exists at least one. So if someone gives you an open cover, that open cover may not contain an open set equal to the space.
Dec
11
revised Do the singular matrices form a topological manifold
deleted 13 characters in body
Dec
11
revised Do the singular matrices form a topological manifold
deleted 13 characters in body
Dec
10
revised Do the singular matrices form a topological manifold
added 46 characters in body
Dec
10
awarded  Promoter
Dec
10
revised Do the singular matrices form a topological manifold
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Dec
9
revised Do the singular matrices form a topological manifold
edited tags
Dec
9
revised Do the singular matrices form a topological manifold
added 324 characters in body
Dec
9
accepted Show that if $z\in\mathbb{E}$ is a fixed point of $\phi\in \operatorname{Aut}(\mathbb{E})$, then so is $\frac{1}{\overline{z}}$
Dec
9
accepted Constructing a (smooth) diffeomorphism between non-smooth manifolds
Dec
8
comment Constructing a (smooth) diffeomorphism between non-smooth manifolds
why can't you talk about the charts $\psi$ and $\phi$ if the atlas isn't smooth, couldn't I still find an appropriate $f$ such that $\psi\circ f\circ \phi^{-1}$ is smooth?
Dec
8
comment Constructing a (smooth) diffeomorphism between non-smooth manifolds
ok this makes sense. I guess my issue was that you can compose non-smooth functions and get a smooth function (for instance composing a non-smooth function with its inverse), could you not cleverly do something like that for $\psi\circ f\circ\phi^{-1}$?
Dec
8
comment Constructing a (smooth) diffeomorphism between non-smooth manifolds
ok great! that really clears things up, thanks! Although I guess my original question would be that you can compose two non-smooth functions and obtain a smooth function, (for example composing a non-smooth function with it's inverse), could you not cleverly do this for the above compositions?
Dec
8
comment Constructing a (smooth) diffeomorphism between non-smooth manifolds
ok so the reason you need a smooth structure is because when you say $f$ and $f^{-1}$ are smooth, what you're really saying is that $\phi_2 \circ f\circ\phi_1^{-1}$ and $\phi_1 \circ f^{-1}\circ\phi_2^{-1}$ are smooth, for appropriate charts? Because otherwise I don't see how you talk about a differentiable function between topological spaces.
Dec
8
comment Constructing a (smooth) diffeomorphism between non-smooth manifolds
I guess maybe I'm confused about diffeomorphisms because it's required that they be smooth, yet what does it mean for this map to be differentiable at all if you're mapping between topological spaces, since you need more than just a topology to define a derivative, does it mean that you compose it with an acceptable chart from each manifold like so: $\phi_2\circ F\circ\phi_1^{-1}$?
Dec
8
comment Constructing a (smooth) diffeomorphism between non-smooth manifolds
@manthanomen, well between $\mathbb{R}$ and $\mathbb{R}$, I could just define the map to be the identity function, which is smooth, but then on $\mathbb{R}$ I could define some atlas on it which isn't smooth. Maybe I'm not understanding things correctly as differential geometry seems to be notorious for having competing definitions for its standard objects.
Dec
8
asked Constructing a (smooth) diffeomorphism between non-smooth manifolds
Dec
7
accepted Let $R$ be a Dedekind domain. If $I$,$J$ are $R$-ideals, prove that there exists an $\alpha\in I$ such that $\gcd(\langle\alpha\rangle,IJ)=I$