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We consider analytic families of planar vector fields depending analytically on the parameters in Λ\Lambda that guarantee the existence of a (may be degenerate and with characteristic directions) monodromic singularity. We characterize the structure of the asymptotic Dulac series of the Poincaré map associated to the singularity when the family possesses a Puiseux inverse integrating factor in terms of its multiplicity and index. This characterization is only valid in a restricted monodromic parameter space Λ\Λ∗\Lambda \backslash {\Lambda }^{\ast } associated to the nonexistence of local curves with zero angular speed. As a byproduct, we are able to study the center-focus problem (under the assumption of the existence of some Cauchy principal values) in very degenerated cases where no other tools are available. We illustrate the theory with several nontrivial examples.

In this work, we present some criteria about the existence and nonexistence of both Puiseux inverse integrating factors V$V$ and Puiseux first integrals H$H$ for planar analytic vector fields having a monodromic singularity. These functions are a wide generalization of their formal R[[x,y]]$\mathbb {R}[[x,y]]$ or algebraic counterpart in Cartesian coordinates (x,y)$(x,y)$. We prove that none of the functions H$H$ and V$V$ can be used to characterize degenerate centers although the existence of H$H$ is a sufficient center condition. [ABSTRACT FROM AUTHOR]

This work is focused in the center problem for nilpotent singularities of differential systems in the plane. Although there are involved methods to approach the center problem in this work, we present a new stability criteria based on the existence of the complex separatrices passing through the singular point, and independent of the normal form. The algebraic method needs only the computation of such complex separatrices up to certain order, which can be done with a computer algebra package. [ABSTRACT FROM AUTHOR]

The Riccati polynomial differential systems are differential systems of the form x ′ = c 0 (x) , y ′ = b 0 (x) + b 1 (x) y + b 2 (x) y 2 , where c 0 and b i for i = 0 , 1 , 2 are polynomial functions. We characterize all the Riccati polynomial differential systems having an invariant algebraic curve. We show that the coefficients of the first four highest degree terms of the polynomial in the variable y defining the invariant algebraic curve determine completely the Riccati differential system. A similar result is obtained for any Abel polynomial differential system. [ABSTRACT FROM AUTHOR]

The emergence of a minimal observable length of order of the Planck scale is a prediction of many quantum theories of gravity. However, the question arises as to whether this is a real fundamental length affecting nature in all of its facets, including spacetime. In this work, we show that the quantum measurement process implies the existence of a minimal measurable length and consequently the apparent discretization of spacetime. The obtained result is used to infer the value of zero-point energy in the universe, which is found to be in good agreement with the observed cosmological constant. This potentially offers some hints towards the resolution of the cosmological constant problem. [ABSTRACT FROM AUTHOR]

In this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.

We study the local analytical integrability in a neighborhood of p: ¯q resonant singular point of a two-dimensional vector field and its connection to time-reversibility with respect to the nonsmooth involution (x,y) = (Yp/q, Xq/p): Some generalizations of the theory developed by Sibirsky for the 1: ¯1 resonant case to the p: ¯q resonant case are presented. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of horizons in circular accelerated motions and its consequences. One particular case is the existence of two horizons in any uniform circular motion. The radiation of the Poincaré invariant vacuum is related to the spontaneous breakdown of the conformal symmetry in Quantum Field Theory The main consequence of the existence of these horizons is the Unruh radiation coming from such horizons. This consequence allows us to study the possible experimental detection of the Unruh radiation in such motions. The radiation of the Poincaré invariant vacuum is related to the spontaneous breakdown of the conformal symmetry in Quantum Field Theory. This radiation is associated with an effective temperature that can be detected using an Unruh–DeWitt detector. In fact, this effective temperature at the relativistic limit depends linearly with respect to the proper acceleration. However, in general, this dependence is not linear, contrary of what happens in the classical Unruh effect. In the relativistic limit and high density case, the uniform circular motion becomes a rotating black hole. This allows for future studies of pre-black hole configurations. [ABSTRACT FROM AUTHOR]