arXiv subjects

Fernando Reis

Publications and source records attributed to Fernando Reis.

7 recordsLinked to original sources

arXiv2024

On the topological invariance of the algebraic multiplicity of holomorphic foliations

In this paper, we address one of the most basic and fundamental problems in the theory of foliations and ODEs, the topological invariance of the algebraic multiplicity of a holomorphic foliation. For instance, we prove an adapted version of A'Campo-L\^e's Theorem for foliations, i.e., the algebraic multiplicity equal to one is a topological invariant in dimension two. This result is further generalized to higher dimensions under mild conditions; as a consequence, we prove that saddle-nodes are topologically invariant. We prove that the algebraic multiplicity is a topological invariant in several classes of foliations that contain, for instance, the generalized curves and the foliations of second type. Additionally, we address a fundamental result by Rosas-Bazan, which states that the existence of a homeomorphism extending through a neighborhood of the exceptional divisor of the first blow-up implies the topological invariance of the algebraic multiplicity. We show that the result holds if the homeomorphism extends locally near a singularity, even if it does not extend over the entire divisor.

math.CV↗

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

On flags of holomorphic foliations associated with singular second-order ordinary differential equations

We consider germs of holomorphic vector fields at the origin of $\mathbb{C}^3$, with non-isolated singularities that are tangent to a holomorphic foliation of codimension one. This configuration is known as a $2$-flag of foliations. The focus is on cases where this geometric structure originates from second-order ordinary differential equations. We investigate the behavior of the singular sets associated with the foliations under consideration. Furthermore, we present a classification for second-order equations that admit a $2$-flag of foliations. Finally, we propose a general method for constructing germs of $2$-flags of foliations at the origin of $\mathbb{C}^n$, with suitable properties of the singular sets, and we conclude by demonstrating that under generic assumptions, every equation of order greater than or equal to two is associated formally with a germ of 2-flag of holomorphic foliations at $(\mathbb{C}^3,0)$.

math.DS↗

arXiv2023

A brief introduction on residue theory of holomorphic foliations

This is a survey paper dealing with holomorphic foliations, with emphasis on residue theory and its applications. We start recalling the definition of holomorphic foliations as a subsheaf of the tangent sheaf of a manifold. The theory of Characteristic Classes of vector bundles is approached from this perspective. We define Chern classes of holomorphic foliations using the Chern-Weil theory and we remark that the Baum-Bott residue is a great tool that help us to classify some foliations. We present throughout the survey several recent results and advances in residue theory. We finish by presenting some applications of residues to solve for example the Poincar\'e problem and the existence of minimal sets for foliations.

math.AG↗

arXiv2019

On the integrability of Hill's equation of the motion of the moon

We study under the standpoint of integrable complex analytic 1-forms (complex analytic foliations), a class of second order ordinary differential equations with periodic coefficients. More precisely, we study Hill's equations of motion of the moon, which are related to the dynamics of the system Sun-Earth-Moon. We associate to the {\em complex Hill equation} an integrable complex analytic one-form in dimension three. This defines a {\it Hill foliation}. The existence of first integral for a Hill foliation is then studied. The simple cases correspond to the existence of rational or Liouvillian first integrals. We then prove the existence of a {\it Bessel type} first integral in a more general case. We construct a standard two dimensional model for the foliation which we call {\it Hill fundamental form}. This plane foliation is then studied also under the standpoint of reduction of singularities and existence of first integral. For the more general case of the Hill equation, we prove for the corresponding Hill foliation, the existence of a Laurent-Fourier type formal first integral. Our approach suggests that there may be a class of plane foliations admitting Bessel type first integrals, in connection with the classification of (holonomy) groups of germs of complex diffeomorphisms associate to a certain class of second order ODEs.

math.DS↗

arXiv2018

Assessing the quality of home detection from mobile phone data for official statistics

Mobile phone data are an interesting new data source for official statistics. However, multiple problems and uncertainties need to be solved before these data can inform, support or even become an integral part of statistical production processes. In this paper, we focus on arguably the most important problem hindering the application of mobile phone data in official statistics: detecting home locations. We argue that current efforts to detect home locations suffer from a blind deployment of criteria to define a place of residence and from limited validation possibilities. We support our argument by analysing the performance of five home detection algorithms (HDAs) that have been applied to a large, French, Call Detailed Record (CDR) dataset (~18 million users, 5 months). Our results show that criteria choice in HDAs influences the detection of home locations for up to about 40% of users, that HDAs perform poorly when compared with a validation dataset (the 35{\deg}-gap), and that their performance is sensitive to the time period and the duration of observation. Based on our findings and experiences, we offer several recommendations for official statistics. If adopted, our recommendations would help in ensuring a more reliable use of mobile phone data vis-\`a-vis official statistics.

cs.CY↗

arXiv2018

Detecting home locations from CDR data: introducing spatial uncertainty to the state-of-the-art

Non-continuous location traces inferred from Call Detail Records (CDR) at population scale are increasingly becoming available for research and show great potential for automated detection of meaningful places. Yet, a majority of Home Detection Algorithms (HDAs) suffer from "blind" deployment of criteria to define homes and from limited possibilities for validation. In this paper, we investigate the performance and capabilities of five popular criteria for home detection based on a very large mobile phone dataset from France (~18 million users, 6 months). Furthermore, we construct a data-driven framework to assess the spatial uncertainty related to the application of HDAs. Our findings appropriate spatial uncertainty in HDA and, in extension, for detection of meaningful places. We show how spatial uncertainties on the individuals' level can be assessed in absence of ground truth annotation, how they relate to traditional, high-level validation practices and how they can be used to improve results for, e.g., nation-wide population estimation.

cs.CY↗