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In this paper we prove the implicit function theorem for , and as an interesting application of using blow-analytic homeomorphisms, we describe a very easy way to resolve singularities of analytic curves.

Let $A\subset {ℝ}^n$ be a set-germ at $0\in {ℝ}^n$ such that $0\in \overline{A}$. We say that $r\in S^{n-1}$ is a direction of $A$ at $0\in {ℝ}^n$ if there is a sequence of points $\left\{x_i\right\}\subset A\setminus \left\{0\right\}$ tending to $0\in {ℝ}^n$ such that $\frac{x_i}{\parallel x_i\parallel }\to r$ as $i\to \infty$. Let $D\left(A\right)$ denote the set of all directions of $A$ at $0\in {ℝ}^n$. Let $A,B\subset {ℝ}^n$ be subanalytic set-germs at $0\in {ℝ}^n$ such that $0\in \overline{A}\cap \overline{B}$. We study the problem of whether the dimension of the common direction set, $dim\left(D\right(A)\cap D(B\left)\right)$ is preserved by bi-Lipschitz homeomorphisms. We show that although it is not true in general, it is preserved if the images of $A$ and $B$ are also subanalytic. In particular...

We show that a subanalytic map-germ (Rⁿ,0) → (Rⁿ,0) which is arc-analytic and bi-Lipschitz has an arc-analytic inverse.

In a previous paper by Koike and Paunescu, it was introduced the notion of direction set for a subset of a Euclidean space, and it was shown that the dimension of the common direction set of two subanalytic subsets, called , is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we first prove our main theorem...

In this paper we construct non-trivial examples of isomorphisms and we obtain, via toric modifications, an inverse function theorem in this category. We also show that any analytic curve in ${ℝ}^n,n\ge 3$, can be deformed via a rational blow- analytic isomorphism of ${ℝ}^n$, to a smooth analytic arc.