Search

The aim of this work is to give conditions on the parameters of a family of analytic planar vector fields with a fixed Newton diagram and a monodromic singularity in order to guarantee that the coefficient (generalized first Lyapunov constant) of the linear part of the Poincaré map has an explicit formula that depends only on the Newton diagram. As far as we know, all the monodromic classes appearing in the literature satisfy that formula. [ABSTRACT FROM AUTHOR]

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 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]

A complete algebraic characterization of the first integrals of the Higgins–Selkov, Selkov and Brusellator oscillators is given here. The existence of symmetries sometimes forces the existence of such first integrals. The nonexistence of centers for such oscillators is also proved. In order to determine the Puiseux integrability of such systems, the multiple Puiseux solutions are also studied. [ABSTRACT FROM AUTHOR]

We consider a complex differential system with a weak saddle at the origin and we characterize the existence of a local analytic first integral around the weak saddle. If the system does not have a fixed degree and instead the degree is arbitrarily large, the family can have a numerable infinite number of integrability cases. [ABSTRACT FROM AUTHOR]

The existence of a minimal observable length modifies the Heisenberg's uncertainty principle at Plank scales and leads to some modifications of the Dirac equation. Here, we consider the generalized uncertainty principle (GUP) theory in order to deduce a generalized Dirac equation and solve its eigenvalue problem for a particle within a gravitational field created by a central mass. We use two different approximations to tackle the problem, based on the Schwarzschild and a modified Schwarzschild metrics. [ABSTRACT FROM AUTHOR]

Let F = (f , g) : R 2 → R 2 be a polynomial map such that det ⁡ (D F (x , y)) is nowhere zero and F (0 , 0) = (0 , 0). In this work we give a new sufficient condition for the injectivity of F. We also state a conjecture when det ⁡ (D F (x , y)) = constant ≠0 and F (0 , 0) = (0 , 0) equivalent to the Jacobian conjecture. [ABSTRACT FROM AUTHOR]

We provide the sufficient conditions for the simultaneous existence of centres for two families of planar double-reversible quintic systems. Computing the focal values and using crossed resultants and Gröbner bases, we find the centre conditions for such systems. The results obtained show that there exist a kind of symmetry where the number of centers in a quintic system is more than 5, so higher than the degree of the system. This fact gives a counterexample to the question posed in a previous work. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in how the local cyclicity of a family of centers depends on the parameters. This fact was pointed out in [21] , to prove that there exists a family of cubic centers, labeled by C D 31 12 in [25] , with more local cyclicity than expected. In this family, there is a special center such that at least twelve limit cycles of small amplitude bifurcate from the origin when we perturb it in the cubic polynomial general class. The original proof has some crucial missing points in the arguments that we correct here. We take advantage of a better understanding of the bifurcation phenomenon in nongeneric cases to show two new cubic systems exhibiting 11 limit cycles and another exhibiting 12. Finally, using the same techniques, we study the local cyclicity of holomorphic quartic centers, proving that 21 limit cycles of small amplitude bifurcate from the origin, when we perturb in the class of quartic polynomial vector fields. [ABSTRACT FROM AUTHOR]

The EPR paradox appears when measurement results of some properties of two distantly entangled particles are correlated in a way that cannot be explained classically, and apparently violate locality. The resolution of the paradox depends on one's interpretation of quantum mechanics. Explanations from quantum mechanics remain commonplace today, but they fail to explain the EPR (Einstein, Podolsky and Rosen) paradox totally in a way than can be accepted by the whole community. Here, we present a simple resolution to this paradox in which the uncertainty in the energy of the two-particle system is reduced by its lack of interaction during the journey so that the uncertainty in time becomes greater than the time they have been separating. Consequently, the present and past become indistinguishable because when we measure an observable in the system its value is the same as if the two particle were still together or very close. It is also argued that the destruction of information as the present and past become identical should release heat by Landauer's principle, and this might make this proposal testable. [ABSTRACT FROM AUTHOR]

In this paper, we study conditions under which the zero-set of the inverse Jacobi multiplier of a smooth vector field contains its attractor/repeller compact sets. The work generalizes previous results focusing on sink singularities, orbitally asymptotic limit cycles, and monodromic attractor graphics. Taking different flows on the torus and the sphere as canonical examples of attractor/repeller sets with different topologies, several examples are constructed illustrating the results presented. [ABSTRACT FROM AUTHOR]

The generalization of the double-slit experiment to an arbitrary number of slits is worthwhile to validate the fundamentally probabilistic nature of quantum mechanics and presents an increasing number of applications. In this work we present an explanation of this experiment from the vacuum fluctuations near the slits with two different approaches: the Heisenberg uncertainty principle and the Kubo-Martin-Schwinger relation characterizing thermalized states. Both descriptions reach analogous results when applied to the sea of virtual particles near the slits. [ABSTRACT FROM AUTHOR]

In this paper we show that Modified Inertia, i.e., the modification of inertia predicted by some alternative theories of gravity at cosmic scales, can be naturally derived within the framework of the extended uncertainty principle (EUP). Specifically, we consider two possible extensions of the Heisenberg uncertainty principle (HUP), corresponding to two different deformations of the fundamental commutator: the first one provides the natural generalization of the HUP to the (anti)-de Sitter spacetime and is endowed with only a quadratic correction in the uncertainty position. On the other hand, the second model contains both linear and quadratic extra terms. We prove that modified inertia is a direct consequence of the minimal acceleration experienced by any body due to the cosmic expansion. The obtained results are then discussed in connection with the empirical predictions of Modified Newtonian dynamics (MoND). The requirement of consistency between the two approaches allows us to fix the adjustable constant which marks the transition between the Newtonian and deep-MoND regimes. [ABSTRACT FROM AUTHOR]

The most important observable consequence of the vacuum fluctuations is the Casimir effect. Its classical manifestation is a force between two uncharged conductive plates placed a few nanometers apart. In this work, we improve the deduction of the Casimir effect from the uncertainty principle by using an effective radius for the quantum fluctuations. Moreover, the existence of this effective distance is discussed. Finally, a heuristic derivation of the Casimir energy for a spherical shell and a sphere-plate cases is given. [ABSTRACT FROM AUTHOR]

In this paper, we present a criterion for determining the formal Weierstrass nonintegrability of some polynomial differential systems in the plane ℂ 2 . The criterion uses solutions of the form y = f (x) of the differential system in the plane and their associated cofactors, where f (x) is a formal power series. In particular, the criterion provides the necessary conditions in order that some polynomial differential systems in ℂ 2 would be formal Weierstrass integrable. Inside this class there exist non-Liouvillian integrable systems. Finally we extend the theory of formal Weierstrass integrability to Puiseux Weierstrass integrability. [ABSTRACT FROM AUTHOR]

The Black hole cosmology is a cosmological model in which the observable universe is the interior of a black hole. When this cosmology was proposed, it was unknown that the expansion of the universe is accelerated and therefore the cosmological model should have a contribution of the dark energy. In this work we revisited Black hole cosmology in relation to the currently accepted model of the universe. However this model up to now has insurmountable problems and the unique main contribution of the Black hole cosmology is the cosmological horizon whose existence guarantees the existence of an entropic force that gives the correct accelerated expansion of the universe. The justification about the existence of this cosmological horizon is still open although recent works within the so-called entropic cosmology provide models that fit strongly with observation and that reinforce the idea of the existence of the cosmological horizon. [ABSTRACT FROM AUTHOR]

Recently a criterion has been given for determining the weakly formal Weierstrass non-integrability of polynomial differential systems in C2. Here we extend this criterion for determining the strongly formal Weierstrass non-integrability which includes the weakly formal Weierstrass non-integrability of polynomial differential systems in C2. The criterion is based on the solutions of the form y = f (x) with ∈ (x) 2 C[[x]] of the differential system whose integrability we are studying. The results are applied to a differential system that contains the famous force-free Duffing and the Duffing-Van der Pol oscillators. [ABSTRACT FROM AUTHOR]

We provide a method to compute the necessary conditions of integrability for planar differential systems with a weak saddle at the origin. The method is used to characterize the local analytic integrability for resonant saddles of complex systems of the form x ˙ = p x , and y ˙ = − y + f (y) x , 0 ≠ p ∈ N , with f analytic at the origin and with f (0) = 0. [ABSTRACT FROM AUTHOR]

The double-slit experiment is a demonstration of wave-particle duality and one of the most fundamental experiments that help us understand the nature of quantum mechanics. In this work, we give a new explanation of this experiment in terms of the uncertainty principle and vacuum fluctuations. This explanation allows one to understand why the electron interferes with itself when being shot through the double-slit. [ABSTRACT FROM AUTHOR]

Abstract We deal here with planar analytic systems x ˙ = X (x , ε) which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by T Σ (q , ε) the time needed by a perturbed orbit that starts from q ∈ Σ to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of x ˙ = X (x , 0) is a continuable critical orbit in a family of the form x ˙ = X (x , ε) if, for any q ∈ Γ and any Σ that passes through q , there exists q ε ∈ Σ a critical point of T Σ (⋅ , ε) such that q ε → q as ε → 0. In this work we study this new problem of continuability. In particular we prove that a simple critical periodic orbit of x ˙ = X (x , 0) is a continuable critical orbit in any family of the form x ˙ = X (x , ε). We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center x ˙ = X (x , 0). Highlights • We define the notion of continuable critical orbit, which is a generalization of critical orbit. • We prove that a simple critical periodic orbit is a continuable critical orbit in any perturbed family. • We give sufficient conditions for the existence of a continuable critical orbit of an isochronous center. [ABSTRACT FROM AUTHOR]

In this paper, we characterize all the Liouvillian first integrals of a cubic polynomial differential system that contains the van der Pol and the Duffing oscillators. It is also shown that the centers correspond to the Liouville integrable cases. [ABSTRACT FROM AUTHOR]

We characterize the centers of the Chiellini Hamiltonian Liénard second-order differential equations x' = y, y' = ..ƒ(x)y ... g(x) where g(x) = f(x)(k -α ƒ(1 + α)R f(x)dx) with α k ∈ R. Moreover we study the phase portraits in the Poincare disk of these systems when f(x) is linear. [ABSTRACT FROM AUTHOR]

The quartic-linear polynomial differential systems having at least one finite singularity are affine equivalent to systems of the form where P and Q are coprime, P i are homogeneous polynomials of degree i and (otherwise it is cubic-linear) and Q(x, y) is either x or y. In this paper, we classify all the quartic-linear systems with Q(x, y) = y which have a global C∞ first integral. We use the local characterization of first integrals, partition of unity in and smoothness of first integrals in canonical regions. [ABSTRACT FROM AUTHOR]

Abstract We give a complete algebraic characterization of the first integrals of the Rayleigh–Duffing oscillator. We prove the non existence of centers of such system and we study the form of the singular first integrals at the origin. [ABSTRACT FROM AUTHOR]

The Casimir effect is one of the observable consequences of the vacuum fluctuations. The Casimir effect manifests itself as a force between two uncharged conductive plates in a vacuum placed a few nanometers apart. In this work, we try to deduce the Casimir effect directly from the uncertainty principle. [ABSTRACT FROM AUTHOR]

In non-relativistic quantum mechanics, the Heisenberg Uncertainty Principle states a fundamental limit to the accuracy in the measurement of pairs of conjugate variables, such as position and momentum. Based on a semiclassical geometric approach, it has been recently proposed a generalization of the uncertainty principle under the relativistic case, which could be extended to General Relativity. This formalism was applied to the Schwarzschild and de Sitter spacetime, showing that the uncertainty relations obtained can be mapped into deformations of Generalized Heisenberg principles well-known in the literature and obtained from the different models of quantum gravity proposed. In the present study, the generalized Heisenberg Principle is derived from the commutator relation, and has been applied to the classical gravitational tests and the derived consequences are framed and analyzed. [ABSTRACT FROM AUTHOR]

In this work the algebraic characterization of the Darboux, Liouville, and Puiseux first integrals of the vocal fold model is given. The multiple Puiseux solutions, also defined in Puiseux integrability, are studied for the first time. • The Darboux polynomials of the vocal fold are provided. • The Darboux and Liouville integrability are studied. • The Puiseux integrability is studied. • The multiple Puiseux solutions are studied for the first time. [ABSTRACT FROM AUTHOR]

We study the simultaneous existence of centres for two families of planar Zq-equivariant systems. First, we give a short review about Zq-equivariant systems. Next, we present the necessary and sufficient conditions for the simultaneous existence of centres for a Z2-equivariant cubic system and for a Z2- equivariant quintic system. [ABSTRACT FROM AUTHOR]

The Moon–Rand systems, developed to model control flexible space structures, are systems of differential equations in R 3 of the form u ̇ = v , v ̇ = − u − u w , w ̇ = − λ w + f ( u , v ) . We give a partially positive answer to a recently conjecture for a special class of such systems, called the generalized polynomial Moon–Rand systems in the case when λ ∈ ( 0 , ∞ ) and f is either a homogeneous cubic, quartic, quintic or sextic polynomial. [ABSTRACT FROM AUTHOR]

We give a complete algebraic characterization of the non-degenerated centers for planar trigonometric Liénard systems. The main tools used in our proof are the classical results of Cherkas on planar analytic Liénard systems and the characterization of some subfields of the quotient field of the ring of trigonometric polynomials. Our results are also applied to some particular subfamilies of planar trigonometric Liénard systems. The results obtained are reminiscent of the ones for planar polynomial Liénard systems but the proofs are different. [ABSTRACT FROM AUTHOR]

A modification of inertia (called MiHsC or quantized inertia) has been proposed that assumes that inertia is caused by Unruh radiation, and that this radiation is made inhomogeneous in space by either Rindler horizons caused by acceleration or the distant Hubble horizon. The former predicts the standard inertial mass, and the latter predicts galaxy rotation without dark matter and cosmic acceleration without dark energy. It is proposed here that this model can be derived in an alternative way by assuming that the sum of mass ( M), energy ( E) and the information content of horizons ( I) is conserved ( EMI) so that mass-energy is released in a discrete manner when the area of a Rindler horizon reduces. This model could be tested by looking for the quantization of inertial mass and acceleration at very high accelerations, and may provide an explanation for the cosmological constant problem. [ABSTRACT FROM AUTHOR]

In this paper we study the center problem for Abel polynomial differential equations of second kind. Computing the focal values and using modular arithmetics and Gröbner bases we find the center conditions for such systems for lower degrees. [ABSTRACT FROM AUTHOR]

We study the local analytic integrability for real Liénard systems, x ̇ = y − F ( x ) , y ̇ = x , with F ( 0 ) = 0 but F ′ ( 0 ) ≠ 0 , which implies that it has a strong saddle at the origin. First we prove that this problem is equivalent to study the local analytic integrability of the [ p : − q ] resonant saddles. This result implies that the local analytic integrability of a strong saddle is a hard problem and only partial results can be obtained. Nevertheless this equivalence gives a new method to compute the so-called resonant saddle quantities transforming the [ p : − q ] resonant saddle into a strong saddle. [ABSTRACT FROM AUTHOR]

In this work we study the global analytic integrability, Louville integrability and Puiseux integrability of the Tolman–Oppenheimer–Volkoff equation. [ABSTRACT FROM AUTHOR]

In this paper, we first present a survey of the known results on limit cycles and center conditions for Liénard differential systems. Next we propose a generalization of such systems and we study their center conditions and the number of small-amplitude limit cycles that can bifurcate from the origin. Computing the focal values and using Gröbner bases we find the center conditions for such systems up to a certain degree. We also establish a conjecture about the center conditions for such systems when they have arbitrary degree. [ABSTRACT FROM AUTHOR]

In this paper, we provide necessary and sufficient conditions for the existence of local analytic first integrals for a seventh-parameter family of complex cubic systems called the complex Kukles systems. [ABSTRACT FROM AUTHOR]

We characterize all centers of planar weight-homogeneous polynomial vector fields. Moreover we classify all centers of planar weighthomogeneous polynomial vector fields of degrees 6 and 7. [ABSTRACT FROM AUTHOR]

Various attempts have been made to fully explain the mechanism by which a body has inertial mass. Recently, it has been proposed that this mechanism is as follows: when an object accelerates in one direction, a dynamical Rindler event horizon forms in the opposite direction, suppressing Unruh radiation on that side by a Rindler-scale Casimir effect whereas the radiation on the other side is only slightly reduced by a Hubble-scale Casimir effect. This produces a net Unruh radiation pressure force that always opposes the acceleration, just like inertia, although the masses predicted are twice those expected, see Ref. 17. In a later work, an error was corrected so that its prediction improves to within 26% of the Planck mass, see Ref. 10. In this paper, the expression of the inertial mass of a elementary particle is derived from the holographic scenario giving the exact value of the mass of a Planck particle when it is applied to a Planck particle. [ABSTRACT FROM AUTHOR]

In this article we study the center problem for polynomial BiLiénard systems of degree n. Computing the focal values and using Gröbner bases we find the center conditions for such systems for n = 6. We also establish a conjecture about the center conditions for polynomial BiLiéenard systems of arbitrary degree. [ABSTRACT FROM AUTHOR]

In this paper we study the center problem for polynomial differential systems and we prove that any center of an analytic differential system is analytically reducible. We also study the centers for the Cherkas polynomial differential systems x = y, y = P0(x) + P1(x)y + P2(x)y2, where Pi(x) are polynomials of degree n, P0'(0) = 0 and P0'(0) < 0. Computing the focal values we find the center conditions for such systems for degree 3, and using modular arithmetics for degree 4. Finally we do a conjecture about the center conditions for Cherkas polynomial differential systems of degree n. [ABSTRACT FROM AUTHOR]

Non-extensive Tsallis thermostatistics is a widespread paradigm to describe large-scale gravitational systems. In this work we use Tsallis Cosmology to study thermodynamic gravity and derive modified Friedmann equations. We show that corrections induced by non-extensivity affect the Hubble function evolution during the radiation-dominated epoch. In turn, this leads to non-trivial modifications of the mass density and pressure content of the Universe, which provide a viable mechanism allowing for baryogenesis, even in the presence of the standard interaction between the Ricci scalar and baryon current. By demanding consistency with current observational bounds on baryogenesis, we constrain Tsallis δ parameter to be | δ − 1 | ≃ 10 − 3. Based on the recently established connection between Tsallis thermostatistics and the quantum gravitational generalization of the uncertainty principle at Planck scale (GUP), we finally show that this bound is in agreement with the estimation of the GUP parameter predicted by many quantum gravity models. [ABSTRACT FROM AUTHOR]

Let F = ( f , g ) : R 2 → R 2 be a polynomial map such that det ⁡ D F ( x , y ) is different from zero for all ( x , y ) ∈ R 2 and F ( 0 , 0 ) = ( 0 , 0 ) . We prove that for the injectivity of F it is sufficient to assume that the higher homogeneous terms of the polynomials f f x + g g x and f f y + g g y do not have real linear factors in common. The proofs are based on qualitative theory of dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential systems, and it unifies many different kinds of averaging methods. Applying our theory to autonomous analytic differential systems, we obtain some conditions on the existence of limit cycles and integrability. For polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues, we prove that there always exists a positive number N such that if its first N averaging functions vanish, then all averaging functions vanish, and consequently there exists a neighborhood of the origin filled with periodic orbits. Consequently if all averaging functions vanish, the origin is a center for n = 2 . Furthermore, in a punctured neighborhood of the origin, the system is C ∞ completely integrable for n > 2 provided that each periodic orbit has a trivial holonomy. Finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities. [ABSTRACT FROM AUTHOR]