Splitting of separatrices for rapid degenerate perturbations of the classical pendulum

In this work we study the splitting distance of a rapidly perturbed pendulum $H(x,y,t)=\frac{1}{2}y^2+(\cos(x)-1)+\mu(\cos(x)-1)g\left(\frac{t}{\varepsilon}\right)$ with $g(\tau)=\sum_{|k|>1}g^{[k]}e^{ik\tau}$ a $2\pi$-periodic function and $\mu,\varepsilon \ll 1$. Systems of this kind undergo exponentially small splitting and, when $\mu\ll 1$, it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided $g^{[\pm 1]}\neq 0$. Our study focuses on the case $g^{[\pm 1]}=0$ and it is motivated by two main reasons. On the one hand the general understanding of the splitting, as current results fail for a perturbation as simple as $g(\tau)=\cos(5\tau)+\cos(4\tau)+\cos(3\tau)$. On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency $p/q$ in Arnold's original model for diffusion leads to the consideration of pendulum-like Hamiltonians with $g(\tau)=\sin\left(p\cdot\frac{t}{\varepsilon}\right)+\cos\left(q\cdot\frac{t}{\varepsilon}\right),$ where, for most $p, q\in\mathbb{Z}$ the perturbation satisfies $g^{[\pm 1]}\neq 0$. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton-Jacobi formalism. The leading exponentially small term appears at order $\mu^n$, where $n$ is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it

Synthesis and Photophysical Properties of Cycloplatinated(II) Complexes Bearing Isocyanide and Alkynyl Ligands

Cyclometalated platinum(II) complexes havegained increasing interest during the last yearsdue to their rich chemistry and uniqueluminescent properties, making them suitablefor light emitting diodes (OLEDs),photocatalysis, bioimaging or chemicalsensors. However, there are very scarceantecedents of heteroleptic cycloplatinated(II)compounds with isocyanide and alkynylligands.1To expand this research, we aimed to prepare anew family of luminescent cycloplatinatedcomplexes with 2-(2,4-difluorophenyl)pyridine(dfppy, a) and 4-(2-pyridyl)benzaldehyde(ppy-CHO, b) as cyclometalating ligands. Thealkynyl/tert-butyl isocyanide complexes[Pt(C^N)(CCR)(CNtBu)], bearing 4-ethynylanisole (2) and 2-ethynylthiophene (3)have obtained from the chloride derivatives[Pt(C^N)Cl(CNtBu)] (C^N = dfppy 1a, ppy-CHO 1b) using the Sonogashira protocol.These systems have been designed by takinginto account features that could make thesecomplexes interesting in biological studies.The complexes have been fully characterizedusing NMR spectroscopy together with X-raydiffraction. Their optical properties have beenstudied in detail and interpreted with the helpof TD-DFT calculations. The dfppy derivativesexhibit a mechanochromic behaviour, relatedto their strong tendency to form stackingstructures or aggregates by Pt···Pt and/or ···interactions

Análisis de las actividades de ocio en estudiantes de primer curso de la Facultad de Educación: Posibles repercusiones conductuales asociadas al consumo de alcohol

Based on responses to a questionnaire from a sample of 705 students (mean 19.2 years, 86% female) first year of the Faculty of Education, we investigated possible behavioural effects associated with different levels of alcohol consumption during the weekends (WE). The results showed that with increased alcohol consumption in WE is a concurrent increase in the percentage of smokers, consumers of illegal drugs, complaints of memory and difficulty waking up, and even detect some changes in attitudes and emotional reactions. To emphasize that this profile, with some exceptions related to the intensity, is valid for men and women, and is no related to any previous medical, psychological or school factor.A partir de las respuestas a un cuestionario de una muestra de 705 estudiantes (media 19,2 años, 86% mujeres) de primer curso de la Facultad de Educación, investigamos posibles repercusiones conductuales asociadas a distintos niveles de consumo de alcohol durante el fin de semana (FS). Los resultados mostraron que a mayor consumo de alcohol en FS se produce un aumento concurrente del porcentaje de fumadores, de consumidores de drogas ilegales, de quejas de memoria y de dificultades para despertarse, e incluso se detectan cambios en algunas actitudes y reacciones emocionales. A destacar que este perfil, salvo excepciones asociadas a la intensidad, es válido para varones y mujeres, y que no guarda relación con antecedentes médicos, psicológicos o escolares de los sujetos

Splitting of separatrices for rapid degenerate perturbations of the classical pendulum

In this work we study the splitting distance of a rapidly perturbed pendulum H(x, y, t) = 1 2 y 2 + (cos(x) - 1) + µ(cos(x) - 1)g t e with g(t ) = P |k|>1 g [k] e ikt a 2p-periodic function and µ, e 1. Systems of this kind undergo exponentially small splitting and, when µ 1, it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided g [±1] 6= 0. Our study focuses on the case g [±1] = 0 and it is motivated by two main reasons. On the one hand the general understanding of the splitting, as current results fail for a perturbation as simple as g(t ) = cos(5t ) + cos(4t ) + cos(3t ). On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency p/q in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with g(t ) = sin p · t e + cos q · t e , where, for most p, q ¿ Z the perturbation satisfies g [±1] 6= 0. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton-Jacobi formalism. The leading exponentially small term appears at order µ n , where n is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.Preprin

Olfactory Ensheathing Glia: Drivers of Axonal Regeneration in the Central Nervous System?

Olfactory ensheathing glia (OEG) accompany olfactory growing axons in their entry to the adult mammalian central nervous system (CNS). Due to this special characteristic, considerable attention has been focused on the possibility of using OEG for CNS regeneration. OEG present a large heterogeneity in culture with respect to their cellular morphology and expressed molecules. The specific characteristics of OEG responsible for their regenerative properties have to be defined. These properties probably result from the combination of several factors: molecular composition of the membrane (expressing adhesion molecules as PSA-NCAM, L1 and/or others) combined with their ability to reduce glial scarring and to accompany new growing axons into the host CNS. Their capacity to produce some neurotrophic factors might also account for their ability to produce CNS regeneration