Abstract. The article is about a “desingularization principle” (Theorem 1.14) common to various canonical desingularization algorithms in characteristic zero, and the roles played by the exceptional divisors in the underlying local construction. We compare algorithms of the authors and of Villamayor and his collaborators, distinguishing between the fundamental effect of the way the exceptional divisors are used, and different theorems obtained because of flexibility allowed in the choice of “input data”. We show how the meaning of “invariance ” of the desingularization invariant, and the efficiency of the algorithm depend on the notion of “equivalence”

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Abstract. For Denjoy–Carleman differential function classes C M where the weight sequence M = (Mk) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping

Abstract. This expository article is devoted to the local theory of ultradifferentiable classes of functions, with a special emphasis on the quasianalytic case. Although quasianalytic classes are well-known in harmonic analysis since several decades, their study from the viewpoint of differential analysis and analytic geometry has begun much more recently and, to some extent, has earned them a new interest. Therefore, we focus on contemporary questions closely related to topics in local algebra. We study, in particular, Weierstrass division problems and the role of hyperbolicity, together with properties of ideals of quasianalytic germs. Incidentally, we also present a simplified proof of Carleman’s theorem on the non-surjectivity of the Borel map in the quasianalytic case. 1. Ultradifferentiable function germs 1.1. Historical background. At the end of the nineteenth century [6, 7], Borel produced the first examples of sets E of infinitely differentiable functions on the real line, containing non-analytic functions, and such that any element f in E satisfies the implication (1) (f (j) (0) = 0, j = 0, 1,...) = ⇒ (f = 0). Borel’s examples were typically given by restrictions to the real line of series of rational functions (2) f(z) = ν=1

Abstract. For quasianalytic Denjoy–Carleman differentiable function classes C Q where the weight sequence Q = (Qk) is log-convex, stable under derivations, of moderate growth and also an L-intersection (see (1.6)), we prove the

Abstract. For Denjoy–Carleman differentiable function classes C M where the weight sequence M = (Mk) is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is C M if it maps C M-curves to C M-curves. The category of C M-mappings is cartesian closed in the sense that C M (E, C M (F, G)) ∼ = C M (E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C M-diffeomorphisms is a C M-Lie group but not better. 1.

Abstract. We prove in a uniform way that all Denjoy–Carleman differentiable function classes of Beurling type C (M) and of Roumieu type C {M}, admit a convenient setting if the weight sequence M = (Mk) is log-convex and of moderate growth: For C denoting either C (M) or C {M} , the category of C-mappings is cartesian closed in the sense that C(E, C(F, G)) ∼ = C(E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C-diffeomorphisms is a regular C-Lie group if C ⊇ C ω, but not better. 1.

There are many contexts in analysis and other areas of mathematics where having an explicit and elementary resolution of singularities algorithm is helpful in understanding local properties of real-analytic functions, or proving theorems that depend on local properties of real-analytic functions. In this paper, a geometric classical analysis resolution of

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