Showing papers by "Maria Aparecida Soares Ruas published in 2013"
TL;DR: In this article, the stable singularities of affine λ-equidistants of closed submanifolds of λ = R^q were studied for any pair of nice dimensions.
Abstract: Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine $\lambda$-equidistants of $n$-dimensional closed submanifolds of $\mathbb R^q$, for $q\leq 2n$, whenever $(2n,q)$ is a pair of nice dimensions.
11 citations
01 Sep 2013
TL;DR: In this article, a ring On of holomorphic germs f ∈ On with its power series f(x) = ∑ aαx, where x = x 1 1... x αn n n n. The Milnor number of a germ f with isolated singularity, denoted by μ(f), is algebraically defined as the dimC On/Jf, where Jf denotes the ideal generated by partial derivatives ∂f/∂x1,..,, ∂ f/∆xn and the multiplicity
Abstract: We fix a system of local coordinates x of C. Consider the ring On of holomorphic germs f : (C, 0) → C and denote by mn its maximal ideal. Due to identification between On and the ring of convergent power series C{x1, . . . , xn} we identify a germ f ∈ On with its power series f(x) = ∑ aαx , where x = x1 1 . . . x αn n . The Milnor number of a germ f with isolated singularity, denoted by μ(f), is algebraically defined as the dimC On/Jf , where Jf denotes the ideal generated by partial derivatives ∂f/∂x1, . . ., ∂f/∂xn and the multiplicity m(f) is the lowest degree in the power series expansion of f at 0 ∈ C. A deformation F : (C×C, 0) → (C, 0), F (x, t) = ft(x), of f is μ-constant if μ(ft) = μ(f) for small values of t. We denote by JF = ⟨∂F/∂x1, . . . , ∂F/∂xn⟩, the ideal in On+1 generated by the partial derivatives of F with respect to the variables x1, . . . , xn. The Milnor number is a topological invariant of the singularity, more precisely if two germs of complex hypersurfaces with isolated singularities are homeomorphic, then have the same Milnor number. We also have by [9], for n = 3, that if a family of hypersurfaces
8 citations
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TL;DR: In this paper, a positive answer to Zariski's conjecture for families of singular surfaces with smooth normalization was given. But this conjecture was only for families with a smooth normalisation.
Abstract: We provide a positive answer to Zariski's conjecture for families of singular surfaces in $\mathbb C^3,$ under the condition that the family has a smooth normalisation. As a corollary of the result, we obtain a surprising characterization of the Whitney equisingularity of one parameter families of $\mathcal A$ finitely determined map-germs $f_t: (\mathbb C^2,0) \to (\mathbb C^3,0),$ in terms of the constancy of only one invariant, the Milnor number of the double point locus.
5 citations
TL;DR: In this article, two sufficient conditions are provided for given two K-equivalent map-germs to be bi-Lipschitz A-equivalents, i.e., they can be expressed as follows:
Abstract: In this paper, two sufficient conditions are provided for given two K-equivalent map-germs to be bi-Lipschitz A-equivalent. These are Lipschitz analogues of the known results on C^r-A-equivalence $(0 \leq r \leq \infty)$ for given two K-equivalent map-germs. As a corollary of one of our results, a Lipschitz version of the well-known Fukuda-Fukuda theorem is provided.