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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2009年第7期137卷 2285-2291页

作者： Fischer, Andreas Kurdyka, Krzysztof Univ Saskatchewan Dept Math & Stat Saskatoon SK S7N 5E6 Canada Univ Savoie Math Lab CNRS UMR 5127 F-73376 Le Bourget Du Lac France

We describe rings of subanalytic functions which become continuously differentiable after finitely many local blowings-up with analytic centers.

关键词： subanalytic functions blowings-up arc-analytic function singularities

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On the convergence of the proximal algorithm for nonsmooth functions involving analytic features

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MATHEMATICAL PROGRAMMING 2009年第1-2期116卷 5-16页

作者： Attouch, Hedy Bolte, Jerome Univ Montpellier 2 CNRS I3M UMR 5149 F-34095 Montpellier France Univ Paris 06 Equipe Combinatoire & Optimisat UMR 7090 F-75252 Paris 05 France

We study the convergence of the proximal algorithm applied to nonsmooth functions that satisfy the Lojasiewicz inequality around their generalized critical points. Typical examples of functions complying with these conditions are continuous semialgebraic or subanalytic functions. Following Lojasiewicz's original idea, we prove that any bounded sequence generated by the proximal algorithm converges to some generalized critical point. We also obtain convergence rate results which are related to the flatness of the function by means of Lojasiewicz exponents. Apart from the sharp and elliptic cases which yield finite or geometric convergence, the decay estimates that are derived are of the type O(k(-s)), where s is an element of (0, + infinity) depends on the flatness of the function.

关键词： proximal algorithm Lujasiewicz inequality subanalytic functions

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Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2003年第1期118卷 67-80页

作者： Lin, GH Fukushima, M Kyoto Univ Dept Appl Math & Phys Grad Sch Informat Kyoto 606 Japan Dalian Univ Technol Dept Math Appl Dalian Peoples R China

Recently, some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints were proved by Luo, Pang, and Ralph (Ref. 1). In this paper, we show that those results remain valid under some other mild conditions. One of these conditions, called strong convexity with order sigma, is discussed in detail.

关键词： mathematical programs with equilibrium constraints nonlinear complementarity problems subanalytic sets subanalytic functions Holder continuity strong convexity error bounds

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Lebesgue classes and preparation of real constructible functions

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JOURNAL OF FUNCTIONAL ANALYSIS 2013年第7期264卷 1599-1642页

作者： Cluckers, Raf Miller, Daniel J. Univ Lille 1 CNRS Lab Painleve UMR 8524 F-59655 Villeneuve Dascq France Katholieke Univ Leuven Dept Math B-3001 Heverlee Belgium Emporia State Univ Dept Math Comp Sci & Econ Emporia KS 66801 USA

We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any q > 0 and constructible functions f and mu on E x R-n, we prove a theorem describing the structure of the set {(x, p) is an element of E x (0, infinity]: f (x, .) is an element of L-p(vertical bar mu vertical bar(q)(x))}, where vertical bar mu vertical bar(q)(x) is the positive measure on R-n whose Radon-Nikodym derivative with respect to the Lebesgue measure is vertical bar mu(x, .)vertical bar(q) : y bar right arrow vertical bar mu(x, y)vertical bar(q). We also prove a closely related preparation theorem for f and mu. These results relate analysis (the study of L-p-spaces) with geometry (the study of zero loci). (c) 2013 Elsevier Inc. All rights reserved.

关键词： L-p-spaces Integrability locus Preparation theorem subanalytic functions Constructible functions

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Local convergence of Newton's method for subanalytic variational inclusions

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POSITIVITY 2008年第3期12卷 525-533页

作者： Cabuzel, Catherine Pietrus, Alain Univ Antilles Guyane Lab Anal Dept Math & Informat Campus Fouillole F-97159 Pointe A Pitre Guadeloupe France

This paper concerns variational inclusions of the form 0 is an element of f(x) + F(x) where f is a single locally Lipschitz subanalytic function and F is a set-valued map acting in Banach spaces. We prove the existence and the convergence of a sequence (x(k)) satisfying 0 is an element of f(x(k)) + Delta f(x(k))(x(k+1)- x(k))+ F(x(k+1)) where Delta f(x(k)) lies to partial derivative f(x(k)) which is the Clarke Jacobian of f at the point xk.

关键词： set-valued mapping generalized equations Aubin continuity Lipschitz-like map subanalytic functions directional derivative Newton's method Clarke Jacobian

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Bounding the Decay of Oscillatory Integrals with a Constructible Amplitude Function and a Globally subanalytic Phase Function

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2016年第1期22卷 215-236页

作者： Cluckers, Raf Miller, Daniel J. Univ Lille 1 Lab Painleve CNRS UMR 8524Cite Sci F-59655 Villeneuve Dascq France Katholieke Univ Leuven Dept Wiskunde Celestijnenlaan 200B B-3001 Louvain Belgium Emporia State Univ Dept Math & Econ 1 Kellogg CircleCampus Box 4027 Emporia KS 66801 USA

We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. Our main theorem gives uniform bounds on the decay of parameterized families of oscillatory integrals with a constructible amplitude function and a globally subanalytic phase function, assuming that the amplitude function is integrable and that the phase function satisfies a certain natural condition called the hyperplane condition. As a simple application of this theorem, we also show that any continuous, integrable, constructible function of a single variable has an integrable Fourier transform.

关键词： Oscillatory integrals Fourier transforms subanalytic functions Constructible functions

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Fields with analytic structure

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JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY 2011年第4期13卷 1147-1223页

作者： Cluckers, R. Lipshitz, L. Univ Lille 1 Lab Painleve CNRS UMR 8524 F-59655 Villeneuve Dascq France Katholieke Univ Leuven Dept Math B-3001 Louvain Belgium Purdue Univ Dept Math W Lafayette IN 47907 USA

We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o-minimality is shown. For Henselian valued fields, both the model theory and the analytic theory are developed. We give a list of examples that comprises, to our knowledge, all principal, previously studied, analytic structures on Henselian valued fields, as well as new ones. The b-minimality is shown, as well as other properties useful for motivic integration on valued fields. The paper is reminiscent of papers by Denef and van den Dries [Ann. of Math. 128 (1988)] and by Cohen [Comm. Pure Appl. Math. 22 (1969)], and of the book by Fresnel and van der Put, Rigid Analytic Geometry and its Applications [Birkhauser (2004)], and unifies work by van den Dries, Haskell, Macintyre, Macpherson, Marker, Robinson, and the authors.

关键词： Henselian valued fields o-minimality b-minimality subanalytic functions cell decomposition analytic structure separated power series

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Mellin transforms of power-constructible functions

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ADVANCES IN MATHEMATICS 2024年 459卷

作者： Cluckers, Raf Comte, Georges Rolin, Jean-Philippe Servi, Tamara Univ Lille CNRS UMR 8524 Lab Paul Painleve F-59000 Lille France Katholieke Univ Leuven Dept Math B-3001 Leuven Belgium Univ Savoie Mont Blanc LAMA CNRS UMR 5127 F-73000 Chambery France Univ Bourgogne Inst Math Bourgogne UMR CNRS 5584 F-21000 Dijon France Univ Paris Cite Inst Math Jussieu Paris Rive Gauche CNRS IMJ PRG F-75013 Paris France Sorbonne Univ CNRS IMJ PRG F-75013 Paris France

We consider several systems of algebras of real- and complex- valued functions, which appear in o-minimal geometry and related geometrically tame contexts. For each such system, we prove its stability under parametric integration and we study the asymptotics of the functions as well as the nature of their parametric Mellin transforms. (c) 2024 Published by Elsevier Inc.

关键词： Oscillatory integrals Mellin transform Fourier transform subanalytic functions

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On the holomorphic convexity of reductive Galois coverings over compact Kähler surfaces

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ADVANCES IN MATHEMATICS 2023年第1期431卷

作者： Liu, Yuan Univ Sci & Technol China Inst Geometry & Phys Hefei 230026 Peoples R China

This article generalizes the result of Katzarkov and Ramachandran from algebraic surfaces to Kahler surfaces. We follow their argument to prove the holomorphic convexity of a reductive Galois covering over a compact Kahler surface which does not have two ends, except that we replace the p-adic factorization theorem by an analysis of the singularities of the continuous subanalytic plurisubharmonic exhaustion function.& COPY;2023 Elsevier Inc. All rights reserved.

关键词： Holomorphically convex subanalytic functions Holomorphic foliation

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Approximations in globally subanalytic and Denjoy-Carleman classes

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ADVANCES IN MATHEMATICS 2021年 385卷 107764-107764页

作者： Valette, Anna Valette, Guillaume Uniwersytetu Pedagogicznego Inst Matemat Ul Podchorazych 2 PL-30084 Krakow Poland Uniwersytetu Jagiellonskiego Inst Matemat Ul S Lojasiewicza 6 PL-30348 Krakow Poland

Efroymson's Approximation Theorem asserts that if f is a C-0 semialgebraic mapping on a C-infinity semialgebraic submanifold M of R-n and if epsilon : M -> R is a positive continuous semialgebraic function then there is a C-infinity semialgebraic function g : M -> R such that vertical bar f -g vertical bar < epsilon. We prove a generalization of this result to the globally subanalytic category. Our theorem actually holds in a larger framework since it applies to every function which is definable in a polynomially bounded o-minimal structure (expanding the real field) that admits C-infinity cell decomposition. We also establish approximation theorems for Lipschitz and C-1 definable functions. (C) 2021 Elsevier Inc. All rights reserved.

关键词： subanalytic functions o-minimal structures Analytic functions Nash functions Approximation theorem Efroymson's theorem

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