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2 votes

Reference for the Brown-Gersten property for smooth manifolds

I typed up a proof of this result: Numerable open covers and representability of topological stacks. The result is proved in greater generaility for arbitrary numerable open covers of topological ...
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0 votes

Curvature estimate in Hamilton's Ricci flow paper for traceless $\operatorname{Rm}$ on $4$-dimensional manifold

After a lot more time thinking about it, I think I've figured it out. Let $\Gamma$ be a constant such that $ \Gamma \|g \odot g \| = 1$ (where $\odot$ denotes the Kulkarni Nomizu product, and the only ...
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1 vote

Products of manifolds and locally ringed spaces (over $\mathbb{R}$) coincide?

Yes if you think of manifolds as $C^\infty$-schemes. (Joyces's book on the subject is probably the most accessible reference, but you could try the original paper by Dubuc.) You can see a hint of ...
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1 vote

For a SDE with smooth transition densities, if every point is "path-accessible", is every positive-measure set probabilistically accessible?

By adapting the arguments in Sec. 3.3.6.1 of the Michel & Pardoux notes linked to by Nawaf Bou-Rabee, I think I can prove the result. (I will assume for simplicity that the SDE has global ...
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1 vote

Why do we have sixteen possible configurations of three saddles on one level?

In this portion of the paper, the authors are concerned with the set of level preserving diffeomorphism classes of Morse functions on surfaces with at most three critical points. Index 0/2 critical ...
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1 vote

For a SDE with smooth transition densities, if every point is "path-accessible", is every positive-measure set probabilistically accessible?

The standard approach is to first use Hörmander’s theorem to prove that the law of $Y_t$ has a smooth density with respect to Lebesgue measure. To prove this density is strictly positive, it suffices ...
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12 votes
Accepted

There exists differentiable curves arbitrarily close to the continuous ones

It turns out that something much more general is true and can be found in the literature. Theorem [Thm 3.3, Hirsch, Differential Topology] Let $M$ and $N$ be $C^s$-manifolds (with boundary), $1\le s\...
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4 votes

Reference request: A knot is tame if and only if it has a tubular neighbourhood

The existence of PL tubular neighborhoods, plus a uniqueness statement, may be found in Wall, Locally flat PL submanifolds with codimension two, (Proc. Cambridge Philos. Soc. 63 (1967), 5–8.). Is that ...
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