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Desingularization algorithms, I. Role of exceptional divisors
, 2002
"... 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 ..."
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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”
THE CONVENIENT SETTING FOR NONQUASIANALYTIC DENJOY–CARLEMAN DIFFERENTIABLE MAPPINGS
"... Abstract. For Denjoy–Carleman differential function classes C M where the weight sequence M = (Mk) is logarithmically convex, stable under derivations, and nonquasianalytic of moderate growth, we prove the following: A mapping ..."
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Cited by 5 (5 self)
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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 nonquasianalytic of moderate growth, we prove the following: A mapping
Resolution of singularities
 in Denjoy– Carleman classes, Selecta Math
"... in celebration of Kiju — joy and long life! Contents ..."
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in celebration of Kiju — joy and long life! Contents
ON QUASIANALYTIC LOCAL RINGS
, 2006
"... 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 wellknown in harmonic analysis since several decades, their study from the viewpoint of differential an ..."
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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 wellknown 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 nonsurjectivity 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 nonanalytic 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
THE CONVENIENT SETTING FOR QUASIANALYTIC DENJOY–CARLEMAN DIFFERENTIABLE MAPPINGS
"... Abstract. For quasianalytic Denjoy–Carleman differentiable function classes C Q where the weight sequence Q = (Qk) is logconvex, stable under derivations, of moderate growth and also an Lintersection (see (1.6)), we prove the ..."
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Cited by 1 (1 self)
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Abstract. For quasianalytic Denjoy–Carleman differentiable function classes C Q where the weight sequence Q = (Qk) is logconvex, stable under derivations, of moderate growth and also an Lintersection (see (1.6)), we prove the
THE CONVENIENT SETTING FOR DENJOY–CARLEMAN DIFFERENTIABLE MAPPINGS OF BEURLING AND ROUMIEU TYPE
"... 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 logconvex and of moderate growth: For C denoting either C (M) or C {M} , the category of Cmap ..."
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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 logconvex and of moderate growth: For C denoting either C (M) or C {M} , the category of Cmappings 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 Cdiffeomorphisms is a regular CLie group if C ⊇ C ω, but not better. 1.
THE KOHN ALGORITHM ON DENJOYCARLEMAN CLASSES
, 806
"... Abstract. The equivalence of the Kohn finite ideal type and the D’Angelo finite type with the subellipticity of the ¯ ∂Neumann problem is extended to pseudoconvex domains in C n whose defining function is in a DenjoyCarleman quasianalytic class closed under differentiation. The proof involves alge ..."
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Abstract. The equivalence of the Kohn finite ideal type and the D’Angelo finite type with the subellipticity of the ¯ ∂Neumann problem is extended to pseudoconvex domains in C n whose defining function is in a DenjoyCarleman quasianalytic class closed under differentiation. The proof involves algebraic geometry over a ring intermediate between the ring of realanalytic functions C ω and the ring of smooth functions C ∞ whose Noetherianity or lack thereof is an open problem. It is also shown that such a ring satisfies the √ acc property, one of the strongest
J. Functional Analysis 256 (2009), 35103544 THE CONVENIENT SETTING FOR NONQUASIANALYTIC DENJOY–CARLEMAN DIFFERENTIABLE MAPPINGS
"... Abstract. For Denjoy–Carleman differentiable function classes C M where the weight sequence M = (Mk) is logarithmically convex, stable under derivations, and nonquasianalytic of moderate growth, we prove the following: A mapping is C M if it maps C Mcurves to C Mcurves. The category of C Mmappin ..."
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Abstract. For Denjoy–Carleman differentiable function classes C M where the weight sequence M = (Mk) is logarithmically convex, stable under derivations, and nonquasianalytic of moderate growth, we prove the following: A mapping is C M if it maps C Mcurves to C Mcurves. The category of C Mmappings 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 Mdiffeomorphisms is a C MLie group but not better. 1.
An Elementary CoordinateDependent Local Resolution of Singularities and Applications
, 2008
"... 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 realanalytic functions, or proving theorems that depend on local properties of realanalytic functions. In ..."
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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 realanalytic functions, or proving theorems that depend on local properties of realanalytic functions. In this paper, a geometric classical analysis resolution of