From circular paths to elliptic orbits: A geometric approach to Kepler's motion

The hodograph, i.e. the path traced by a body in velocity space, was introduced by Hamilton in 1846 as an alternative for studying certain dynamical problems. The hodograph of the Kepler problem was then investigated and shown to be a circle, it was next used to investigate some other properties of the motion. We here propose a new method for tracing the hodograph and the corresponding configuration space orbit in Kepler's problem starting from the initial conditions given and trying to use no more than the methods of synthetic geometry in a sort of Newtonian approach. All of our geometric constructions require straight edge and compass only.Comment: 9 pages, 4 figure

Bifunctional oxygen electrocatalysts based on non-critical raw materials: carbon nanostructures and iron-doped manganese oxide nanowires

Alkaline metal-air batteries are unique systems for energy storage. These devices require a bifunctional catalyst in the positive electrode that must perform both the oxygen evolution and reduction reactions (OER and ORR, respectively). Generally, cobalt-based oxides are employed as air electrodes; however, cobalt is a critical raw material. Future battery devices will mandatorily need non-critical raw materials based on highly abundant metals. Here we investigate the feasibility of iron-doped manganese oxide in the form of nanowires (Fe-MONW) combined with carbon nanofibers. MnO2 is known for being active for the ORR, however its activity towards the OER is not yet fully understood. Carbon nanofibers (CNF) on the other hand, provide the necessary electrical conductivity to the catalytic system. Simple methods and economic materials are employed to synthesize the Fe-MONW/CNF composites. Our results show that there is a synergistic effect between CNF and MONW, especially for the ORR, which manifests in an increase in the number of exchanged electrons– from 2.9 to 3.5 – and a shift in the onset potential of 70 mV. Doping MONW with iron further enhances the catalytic activity, for both the ORR and OER. Fe ions generate defects in the manganese oxide structure, favoring the adsorption of oxygen and eventually enhancing the catalytic activity. Fe-doped-MONW shows onset potentials for OER comparable to the benchmark catalyst, IrO2. The improvement on the catalytic activity is particularly evident in terms of the reversibility gap, ΔE. ΔE is the difference between the potential when the current density is 10 mA cm−2 in OER and the half-wave potential for the ORR, being a fundamental parameter to assess the performance of metal-air batteries. The reversibility gap for the best catalyst, 5Fe-MONW/CNF, is ΔE = 922 mV (140 mV lower than non-doped MONW/CNF and between 160 and 320 mV lower than the individual components, MONW and CNF). Endurance tests show remarkable stability of the iron-doped MONW/CNF, with a stable potential and an even lower ΔE of 800 mV for ca. 20 h of operation (charge-discharge cycles at ± 10 mA cm−2)

On the numerical computation of Diophantine rotation numbers of analytic circle maps

In this paper we present a numerical method to compute Diophantine rotation numbers of circle maps with high accuracy. We mainly focus on analytic circle diffeomorphisms, but the method also works in the case of (enough) finite differentiability. The keystone of the method is that, under these conditions, the map is conjugate to a rigid rotation of the circle. Moreover, albeit it is not fully justified by our construction, the method turns to be quite efficient for computing rational rotation numbers. We discuss the method through several numerical examples

Asymptotic behaviour of the domain of analyticity of invariant curves of the standard map

In this paper we consider the standard map, and we study the invariant curve obtained by analytical continuation, with respect to the perturbative parameter E, of the invariant circle of rotation number the golden mean corresponding to the case E=0. We show that, if we consider the parameterization that conjugates the dynamics of this curve to an irrational rotation, the domain of definition of this conjugation has an asymptotic boundary of analyticity when E->0 (in the sense of the singular perturbation theory). This boundary is obtained studying the conjugation problem for the so-called semi-standard map. To prove this result we have used KAM-like methods adapted to the framework of singular perturbation theory, as well as matching techniques to join di erent pieces of the conjugation, obtained in different parts of its domain of analyticity

Asymptotic size of Herman rings of the complex standard family by quantitative quasiconformal surgery

In this paper we consider the complexification of the Arnold standard family of circle maps given by $\widetilde F_{\alpha,\epsilon}(u)=ue^{i\alpha} e^{({\epsilon}/{2}) (u-{1}/{u})}$, with $\alpha=\alpha(\epsilon)$ chosen so that $\widetilde F_{\alpha(\epsilon),\epsilon}$ restricted to the unit circle has a prefixed rotation number $\theta$ belonging to the set of Brjuno numbers. In this case, it is known that $\widetilde F_{\alpha(\epsilon),\epsilon}$ is analytically linearizable if $\epsilon$ is small enough and so it has a Herman ring $\widetilde U_{\epsilon}$ around the unit circle. Using Yoccoz's estimates, one has that the size$\widetilde R_\epsilon$ of $\widetilde U_{\epsilon}$ (so that $\widetilde U_{\epsilon}$ is conformally equivalent to $\{u\in{\mathbb C}: 1/\widetilde R_\epsilon < |u| < \widetilde R_\epsilon\}$) goes to infinity as $\epsilon\to 0$, but one may ask for its asymptotic behavior. We prove that $\widetilde R_\epsilon=({2}/{\epsilon})(R_0+\mathcal{O}(\epsilon\log\epsilon))$, where R0 is the conformal radius of the Siegel disk of the complex semistandard map $G(z)=ze^{i\omega}e^z$, where $\omega= 2\pi\theta$. In the proof we use a very explicit quasiconformal surgery construction to relate $\widetilde F_{\alpha(\epsilon),\epsilon}$ and G, and hyperbolic geometry to obtain the quantitative result

Cooperative Learning in the Implementation of Teaching Chemistry (Didactic Instrumentation) in Engineering in México

AbstractIn engineering you think of chemistry as a difficult and boring subject. Some professors who are teaching it, have seen apathy and lack of interest in the students. This paper presents some findings of an investigation done which allowed to listen to the voice of 250 Mexican students regarding the usefulness of cooperative learning in chemistry. An exploratory-descriptive methodology was applied, together with a pretest and a post-test. By implementing the Cooperative Learning, the importance of the positive interdependence for critical thinking was appraised and a move away was detected from the theoretical content and meaning of the everyday context of the students

Modeling COVID-19 with Uncertainty in Granada, Spain. Intra-Hospitalary Circuit and Expectations over the Next Months

Mathematical models have been remarkable tools for knowing in advance the appropriate time to enforce population restrictions and distribute hospital resources. Here, we present a mathematical Susceptible-Exposed-Infectious-Recovered (SEIR) model to study the transmission dynamics of COVID-19 in Granada, Spain, taking into account the uncertainty of the phenomenon. In the model, the patients moving throughout the hospital’s departments (intra-hospitalary circuit) are considered in order to help to optimize the use of a hospital’s resources in the future. Two main seasons, September–April (autumn-winter) and May–August (summer), where the hospital pressure is significantly different, have been included. The model is calibrated and validated with data obtained from the hospitals in Granada. Possible future scenarios have been simulated. The model is able to capture the history of the pandemic in Granada. It provides predictions about the intra-hospitalary COVID-19 circuit over time and shows that the number of infected is expected to decline continuously from May without an increase next autumn–winter if population measures continue to be satisfied. The model strongly suggests that the number of infected cases will reduce rapidly with aggressive vaccination policies. The proposed study is being used in Granada to design public health policies and perform wise re-distribution of hospital resources in advance.Spanish Ministerio de Economía, Industria y Competitividad (MINECO)Agencia Estatal de Investigación (AEI)Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-PEuropean Union through the Operational Program of the [European Regional Development Fund (ERDF)/European Social Fund (ESF)] of the Valencian Community 2014–2020Ramón Areces Foundation, Madrid, Spain (CIVP18A3920)

Generalized analytical results on n-ejection–collision orbits in the RTBP: analysis of bifurcations

In the planar circular restricted three-body problem and for any value of the mass parameter µ¿(0,1) and n=1 , we prove the existence of four families of n-ejection–collision (n-EC) orbits, that is, orbits where the particle ejects from a primary, reaches n maxima in the (Euclidean) distance with respect to it and finally collides with the primary. Such EC orbits have a value of the Jacobi constant of the form C=3µ+Ln2/3(1-µ)2/3 , where L>0 is big enough but independent of µ and n. In order to prove this optimal result, we consider Levi-Civita’s transformation to regularize the collision with one primary and a perturbative approach using an ad hoc small parameter once a suitable scale in the configuration plane and time has previously been applied. This result improves a previous work where the existence of the n-EC orbits was stated when the mass parameter µ>0 was small enough. Moreover, for decreasing values of C, there appear some bifurcations which are first numerically investigated and afterward explicit expressions for the approximation of the bifurcation values of C are discussed. Finally, a detailed analysis of the existence of n-EC orbits when µ¿1 is also described. In a natural way, Hill’s problem shows up. For this problem, we prove an analytical result on the existence of four families of n-EC orbits, and numerically, we describe them as well as the appearing bifurcations.Peer ReviewedPostprint (author's final draft

Estación de ensayos para la caracterización de celdas de combustible de membrana de intercambio protónico con alimentación de H2 (Monocelda) con carga electrónica integrada

Estación de ensayos para la caracterización de celdas de combustible de membrana de intercambio protónico con alimentación de H2 (Monocelda) con carga electrónica integrada. Estación de Ensayos Para la Caracterización de Celdas de Combustible de Membrana de Intercambio Protónico (PEMFC) con alimentación de H2 (Monocelda) con Carga Electrónica integrada es un sistema para la gestión de gases en una PEMFC que simplificar el control, integrando componentes industriales de alta fiabilidad y robustez. Un sistema desarrollado para facilitar el procesamiento de la información, dotado de una arquitectura de medida y control de carácter innovador que permite el uso de dispositivos de elevadas prestaciones a coste competitivo frente a los equipos de laboratorio similares existentes en el mercado, con una arquitectura de tratamiento de datos compuesta por un procesador central y 4 subsistemas.Peer reviewedConsejo Superior de Investigaciones Científicas (España)A1 Solicitud de patente con informe sobre el estado de la técnic

Pharmacological Profile of the Purinergic P2Y Receptors That Modulate, in Response to ADPβS, the Vasodepressor Sensory CGRPergic Outflow in Pithed Rats

Calcitonin gene-related peptide (CGRP), an endogenous neuropeptide released from perivascular sensory nerves, exerts a powerful vasodilatation. Interestingly, adenosine triphosphate (ATP) stimulates the release of CGRP by activation of prejunctional P2X2/3 receptors, and adenosine 5′-O-2-thiodiphosphate (ADPβS), a stable adenosine diphosphate (ADP) analogue, produces vasodilator/vasodepressor responses by endothelial P2Y1 receptors. Since the role of ADP in the prejunctional modulation of the vasodepressor sensory CGRPergic drive and the receptors involved remain unknown, this study investigated whether ADPβS inhibits this CGRPergic drive. Accordingly, 132 male Wistar rats were pithed and subsequently divided into two sets. In set 1, ADPβS (5.6 and 10 µg/kg·min) inhibited the vasodepressor CGRPergic responses by electrical stimulation of the spinal T9–T12 segment. This inhibition by ADPβS (5.6 µg/kg·min) was reverted after i.v. administration of the purinergic antagonists MRS2500 (300 µg/kg; P2Y1) or MRS2211 (3000 µg/kg; P2Y13), but not by PSB0739 (300 µg/kg; P2Y12), MRS2211 (1000 µg/kg; P2Y13) or the KATP blocker glibenclamide (20 mg/kg). In set 2, ADPβS (5.6 µg/kg·min) failed to modify the vasodepressor responses to exogenous α-CGRP. These results suggest that ADPβS inhibits CGRP release in perivascular sensory nerves. This inhibition, apparently unrelated to activation of ATP-sensitive K+ channels, involves P2Y1 and probably P2Y13, but not P2Y12 receptors.</p