arXiv subjects

Maico Ribeiro

Publications and source records attributed to Maico Ribeiro.

4 recordsLinked to original sources

arXiv2026

Milnor fibrations on rho-tubes and rho-spheres for real analytic map germs

For a real analytic map germ $G:(\mathbb{R}^m,0)\to(\mathbb{R}^p,0)$, we study Milnor fibrations obtained by replacing the squared Euclidean distance with an analytic control function $ρ$ defining the origin. We formulate the corresponding Milnor set and condition (b), derive criteria for tube and sphere fibrations, and examine their dependence on $ρ$. The family $G_ρ=(xz,\,yzρ)$ provides explicit changes of regularity under elliptic controls. In particular, the germ $G(x,y,z)=(xz,\,yz(10x^2+y^2+3z^2))$ admits neither the induced tube nor sphere fibration for the Euclidean control, while both fibrations exist for the adapted function $ρ=10x^2+y^2+3z^2$. Thus the control function can be essential to the local fibration structure.

math.DG↗

arXiv2023

Universally injective and integral contractions on relative Lipschitz saturation of algebras

In this work, we obtain contraction results for a class of diagrams of ring morphisms which strictly includes the ones obtained by Lipman. Further, we present some applications on quotient and in the changing of the base ring in the saturation.

math.AC↗

arXiv2023

Topology of first integrals via Milnor fibrations II

This survey is the continuation of a series of works aimed at applying tools from Singularity Theory to Differential Equations. More precisely, we utilize the powerfull Milnor's Fibration Theory to give geometric-topological classifications of first integrals of differential systems. In the previous paper, systems of first-order quasilinear partial differential equations were examined, focusing on the case of an isolated singularity. Now, we address both cases of isolated and \textit{non-isolated singularities} for more general dynamical systems (namely, \textit{foliations}) that admit at least one first integral. For this, we utilize recently established connections between harmonic morphisms and Milnor fibrations to provide topological and geometric descriptions of the foliations under consideration. In particular, we apply these results to analyze the graph of solutions of some quasilinear systems.

math.DS↗

arXiv2023

Some remarks about $ρ$-regularity for real analytic maps

In this paper, we discuss the concept of $ρ$-regularity of analytic map germs and its close relationship with the existence of locally trivial smooth fibrations, known as the Milnor fibrations. The presence of a Thom regular stratification or the Milnor condition (b) at the origin, indicates the transversality of the fibers of the map G with respect to the levels of a function $ρ$, which guarantees $ρ$-regularity. Consequently, both conditions are crucial for the presence of open book structures and the Milnor fibrations. The work aims to provide a comprehensive overview of the main results concerning the existence of Thom regular stratifications and the Milnor condition (b) for germs of analytic maps. It presents strategies and criteria to identify and ensure these regularity conditions and discusses situations where they may not be satisfied. The goal is to understand the presence and limitations of these conditions in various contexts.

math.DG↗