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Let $M$ be a smooth manifold, $p\in M$ and $f\in C^\infty(M\setminus\{p\})$. We will say that $f$ has a power-law singularity at $p$ of order $\eta$ if for every smooth immersion $\gamma:(-1,1)\to M$ ...

Lets be in the category $\mathbf{M}$ of smooth finite dimensional manifolds with smooth maps: Suppose we have the set of all smooth maps $Hom_\mathbf{M}(R^n,M)$ from $R^n$ to a smooth manifold $M$. ...