C 0 and bi-Lipschitz {mathcal{K}} -equivalence of mappings |
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Authors: | Maria Aparecida Soares Ruas Guillaume Valette |
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Affiliation: | 1. Departamento de Matem??tica, Instituto de Ci??ncias, Matem??ticas e de Computa??o, Universidade de S?o Paulo, Campus de S?o Carlos, Caixa Postal 668, S?o Carlos, SP, 13560-970, Brazil 2. Instytut Matematyczny PAN, ul. ?w. Tomasza 30, 31-027, Krak??w, Poland
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Abstract: | In this paper we investigate the classification of mappings up to ${mathcal{K}}$ -equivalence. We give several results of this type. We study semialgebraic deformations up to semialgebraic C 0 ${mathcal{K}}$ -equivalence and bi-Lipschitz ${mathcal{K}}$ -equivalence. We give an algebraic criterion for bi-Lipschitz ${mathcal{K}}$ -triviality in terms of semi-integral closure (Theorem 3.5). We also give a new proof of a result of Nishimura: we show that two germs of smooth mappings ${f, g: mathbb{R}^n to mathbb{R}^n}$ , finitely determined with respect to ${mathcal{K}}$ -equivalence are C 0- ${mathcal{K}}$ -equivalent if and only if they have the same degree in absolute value. |
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