Equivalence of differentiable mappings and analytic mappings
Nash Manifolds
Triangulation of subanalytic sets
Classification of Nash manifolds
A semi-algebraic analytic manifold and a semi-algebraic analytic map are called a Nash manifold and a Nash map respectively. We clarify the category of Nash manifolds and Nash maps.
Whitney triangulations of semialgebraic sets
A compact semialgebraic set admits a semialgebraic triangulation such that the family of open simplexes forms a Whitney stratification and is compatible with a finite number of given semialgebraic subsets.
Finiteness of semialgebraic types of polynomial functions.
Nash triviality in families of Nash manifolds.
Nash functions on noncompact Nash manifolds
Modified Nash triviality of a family of zero-sets of real polynomial mappings
In this paper we introduce the notion of modified Nash triviality for a family of zero sets of real polynomial map-germs as a desirable one. We first give a Nash isotopy lemma which is a useful tool to show triviality. Then, using it, we prove two types of modified Nash triviality theorem and a finite classification theorem for this triviality. These theorems strengthen similar topological results.
On global Nash functions
Separation, factorization and finite sheaves on Nash manifolds
Global problems on Nash functions.
This is a survey on the history of and the solutions to the basic global problems on Nash functions, which have been only recently solved, namely: separation, extension, global equations, Artin-Mazur description and idempotency, also noetherianness. We discuss all of them in the various possible contexts, from manifolds over the reals to real spectra of arbitrary commutative rings.
Directional properties of sets definable in o-minimal structures
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...
Clarke critical values of subanalytic Lipschitz continuous functions
The main result of this note asserts that for any subanalytic locally Lipschitz function the set of its Clarke critical values is locally finite. The proof relies on Pawłucki's extension of the Puiseux lemma. In the last section we give an example of a continuous subanalytic function which is not constant on a segment of "broadly critical" points, that is, points for which we can find arbitrarily short convex combinations of gradients at nearby points.
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