Masahiro Shiota

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We establish the following result: if the graph of a (nonsmooth) real-extended-valued function f : Rn → R∪{+∞} is closed and admits a Whitney stratification, then the norm of the gradient of f at x ∈ dom f relative to the stratum containing x bounds from below all norms of Clarke subgradients of f at x. As a consequence, we obtain some Morse-Sard type(More)
Proefschrift voorgelegd tot het behalen van de graad van doctor in de wetenschappen, richting informatica aan de transnationale Universiteit Limburg te verdedigen door Preface First of all, I would like to thank Jan Van den Bussche for his support during the last four years. His never ending enthusiasm and advice made this dissertation possible. His(More)
We study the analytic equivalence of real analytic function germs after desingularization and state the cardinality of the classes under this equivalence relation. We consider also the Nash case, and compare these equivalences with the blow-analytic (respectively blow-Nash) equivalence. We prove an approximation result after desingularization: Nash function(More)
By Hironaka Desingularization Theorem, any real analytic function has only normal crossing singularities after a modification. We focus on the analytic equivalence of such functions with only normal crossing singularities. We prove that for such functions C∞ right equivalence implies analytic equivalence. We prove moreover that the cardinality of the set of(More)
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(More)
The Euler characteristic of the link of a real algebraic variety is an interesting topological invariant in order to discuss local topological properties. We prove in the paper that an invariant stronger than the Euler Characteristic is well defined for the link of an algebraic variety: its virtual Poincaré polynomial. Introduction Let X ⊂ R be a real(More)
More than half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word “generically”, this statement is ostensibly true and is nowadays(More)
Approximation of real analytic functions by Nash functions is a classical topic in real geometry. In this paper, we focus on the Nash approximation of an analytic desingularization of a Nash function germ obtained by a sequence of blowings-up along smooth analytic centers. We apply the result to prove that Nash function germs that are analytically(More)
A blow-analytic homeomorphism is an arc-analytic subanalytic homeomorphism, and therefore it induces a bijective mapping between spaces of analytic arcs. We tackle the question of the continuity of this induced mapping between the spaces of arcs, giving a positive and a negative answer depending of the topology involved. We generalise the result to spaces(More)