Some Applications of the Extended Bendixson-Dulac Theorem

During the last years the authors have studied the number of limit cycles of several families of planar vector fields. The common tool has been the use of an extended version of the celebrated Bendixson-Dulac Theorem. The aim of this work is to present an unified approach of some of these results, together with their corresponding proofs. We also provide several applications.Comment: 19 pages, 3 figure

Aplicación del algoritmo de Kalman al estudio de la ritmicidad ultradiana en el comportamiento expontáneo de sujetos humanos aislados

Este trabajo explora la posible existencia de ritmos ultradianos (período comprendido entre 30 minutos y 20 horas) en el comportamiento espontáneo vigil del hombre, analizando espectralmente las señales con el filtra do adaptativo de Kalman. En cuatro sujetos, cada uno de los cuales permaneció aislado en una habitación, se registraron los siguientes comportamientos: deambulación. exploración, autocontacto y alimentación. De forma simultánea se contabilizó la motilidad empleando un procedimiento telemétrico. La información se totalizó de manera acumulativa a intervalos de 5 minutos y se analizó utilizando el algoritmo de Kalman. El método de estimación espectral empleado obedece a un sistema adaptativo basado en predicción lineal, dando lugar a un cálculo recursivo de los parámetros del filtro predictor. La optimización de este procedimiento se pone en evidencia al compararlo con el de Widrow, en el que la ganancia permanece en un valor previamente fijado (~ constante). Todos los tipos de comportamiento (excepto la alimentaciÓn) muestran picos destacados en la banda de frecuencia de 10-20 ciclos por día (período medio 96 minutos) habitual en muchos otros parámetros biológicos del hombre, asi aomo ritmicidades de periodos más cortos.Peer ReviewedPostprint (published version

Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences.Comment: 25 page

Interaction of cochlin and mechanosensitive channel TREK-1 in trabecular meshwork cells influences the regulation of intraocular pressure.

This work was funded by National Institute of Health Grants R01 EY016112, EY015266, and EY014801 and an unrestricted grant to the University of Miami's Bascom Palmer Eye Institute from Research to Prevent Blindness. Financial support from Fight for Sight is gratefully acknowledged. Funding to XG was provided by Instituto de Salud Carlos III, Spain (FIS PI14/00141 and RETIC RD12/0034/0003) and Generalitat de Catalunya (2014SGR1165). In the eye, intraocular pressure (IOP) is tightly regulated and its persistent increase leads to ocular hypertension and glaucoma. We have previously shown that trabecular meshwork (TM) cells might detect aqueous humor fluid shear stress via interaction of the extracellular matrix (ECM) protein cochlin with the cell surface bound and stretch-activated channel TREK-1. We provide evidence here that interaction between both proteins are involved in IOP regulation. Silencing of TREK-1 in mice prevents the previously demonstrated cochlin-overexpression mediated increase in IOP. Biochemical and electrophysiological experiments demonstrate that high shear stress-induced multimeric cochlin produces a qualitatively different interaction with TREK-1 compared to monomeric cochlin. Physiological concentrations of multimeric but not monomeric cochlin reduce TREK-1 current. Results presented here indicate that the interaction of TREK-1 and cochlin play an important role for maintaining IOP homeostasis

Some results on homoclinic and heteroclinic connections in planar systems

Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $\Phi(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${\Phi(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/45294865}n^{3/2}- {2352961656/11108339166925} n^2+O(n^{5/2})$. We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions

The third order Melnikov function of a quadratic center under quadratic perturbation

We study quadratic perturbations of the integrable system (1+x)dH; where H =(x²+y²)=2: We prove that the first three Melnikov functions associated to the perturbed system give rise at most to three limit cycles

Purinergic receptors in ocular inflammation

Inflammation is a complex process that implies the interaction between cells and molecular mediators, which, when not properly 'tuned,' can lead to disease. When inflammation affects the eye, it can produce severe disorders affecting the superficial and internal parts of the visual organ. The nucleoside adenosine and nucleotides including adenine mononucleotides like ADP and ATP and dinucleotides such as P(1),P(4)-diadenosine tetraphosphate (Ap4A), and P(1),P(5)-diadenosine pentaphosphate (Ap5A) are present in different ocular locations and therefore they may contribute/modulate inflammatory processes. Adenosine receptors, in particular A2A adenosine receptors, present anti-inflammatory action in acute and chronic retinal inflammation. Regarding the A3 receptor, selective agonists like N(6)-(3-iodobenzyl)-5'-N-methylcarboxamidoadenosine (CF101) have been used for the treatment of inflammatory ophthalmic diseases such as dry eye and uveoretinitis. Sideways, diverse stimuli (sensory stimulation, large intraocular pressure increases) can produce a release of ATP from ocular sensory innervation or after injury to ocular tissues. Then, ATP will activate purinergic P2 receptors present in sensory nerve endings, the iris, the ciliary body, or other tissues surrounding the anterior chamber of the eye to produce uveitis/endophthalmitis. In summary, adenosine and nucleotides can activate receptors in ocular structures susceptible to suffer from inflammatory processes. This involvement suggests the possible use of purinergic agonists and antagonists as therapeutic targets for ocular inflammation

Some properties of the k-dimensional Lyness' map

This paper is devoted to study some properties of the k-dimensional Lyness' map. Our main result presentes a rational vector field that gives a Lie symmetry for F. This vector field is used, for k less or equal to 5 to give information about the nature of the invariant sets under F. When k is odd, we also present a new (as far as we know) first integral for F^2 which allows to deduce in a very simple way several properties of the dynamical system generated by F. In particular for this case we prove that, except on a given codimension one algebraic set, none of the positive initial conditions can be a periodic point of odd period.Comment: 22 pages; 3 figure

Preprint arXiv: 2209.11253 Submitted on 22 Sep 2022

We study the interplay between symmetry representations of the physical and virtual space on the class of tensor network states for critical spins systems known as field tensor network states (fTNS). These are by construction infinite dimensional tensor networks whose virtual space is described by a conformal field theory (CFT). We can represent a symmetry on the physical index as a commutator with the corresponding CFT current on the virtual space. By then studying this virtual space representation we can learn about the critical symmetry protected topological properties of the state, akin to the classification of symmetry protected topological order for matrix product states. We use this to analytically derive the critical symmetry protected topological properties of the two ground states of the Majumdar-Ghosh point with respect to the previously defined symmetries