New geometrical and dynamical techniques for problems in celestial mechanics

In this thesis, we study the application of symplectic geometry, regular and singular, to symplectic dynamical systems.We start with a motivating case: the relation between symplectic foliations and global transverse Poincaré sections, showing that meaningful dynamical information can be gleaned by simple observations on the geometry of the phase space - in this case, the existence of a symplectic foliation on a hypersurface of the phase space. We then go on study dynamical systems of particular importance in geometry - those given by a group action on a manifold. In particular, we consider a singular symplectic manifold (specifically, a manifold equipped with a symplectic form which blows up in a controlled manner on a hypersurface of that manifold, namely, a b-symplectic form) with a group action preserving the geometry and give a b-symplectic slice theorem which provides an equivariant normal form of the b-symplectic form in the neighbourhood of an orbit. Particular examples of b-symplectic group symmetries are then explored: those given by the cotangent lift of group translation on so-called b-Lie groups. The second part of this thesis focuses on symplectic and b-symplectic dynamical systems coming from celestial mechanics. In particular, the separatrix map of the stable and unstable manifolds of the fixed point at infinity of the planar circular restricted three-body problem is examined and an estimate of the width of the stochastic layer is given - that is the existence of a K.A.M. torus which acts as a boundary to bounded motions is proved. Due to the delicate nature of the problem - namely issues coming from the parabolic nature of the fixed point and exponentially small nature of the splitting, careful control of the errors of the separatrix map is paramount. This is achieved by employing geometric methods, namely, by taking full advantage of generating functions which exist by virtue of the symplectic nature of the system. Finally, motivated by the important role of symplectic geometry in the systems of celestial mechanics in mind, we give examples of degenerate and singular symplectic structures occurring in systems of celestial mechanics which cannot be equipped with a symplectic form.En esta tesis, estudiamos una aplicación de la geometría simpléctica, regular y singular, a sistemas dinámicos simplécticos. Comenzamos con un caso motivador: la relación entre foliaciones simplécticas y secciones transversales de Poincaré globales, que muestra que se puede obtener información significativa de la dinámica mediante simples observaciones sobre la geometría del espacio de fase, en este caso, la existencia de una foliación simpléctica en una hipersuperficie del espacio de fase. Luego continuamos estudiando sistemas dinámicos de particular importancia en geometría, dados por una acción de grupo sobre una variedad. En particular, consideramos una variedad simpléctica singular (específicamente, una variedad equipada con una forma simpléctica que explota de manera controlada en una hipersuperficie de esa variedad, es decir, una forma $b$-simpléctica ) con una acción de grupo que preserva la geometría y damos un teorema de rebanada $b$-simpléctico que proporciona una forma normal equivariante de la forma b-simpléctica en una vecindad de una órbita. Luego se exploran ejemplos particulares de simetrías de grupo b-simplécticas: las dadas por el levantamiento cotangente de la translación (grupal) en los llamados b-grupos de Lie. La segunda parte de esta tesis se enfoca en sistemas dinámicos simplécticos y b-simplécticos provenientes de la mecánica celeste. En particular, se examina el "separatrix map" de variedades estables e inestables del punto fijo al infinito del problema de los tres cuerpos restringido circular plano, y se da una estimación del ancho de la capa estocástica, es decir, la existencia de un toro K.A.M. que actúa como frontera para movimientos acotados. Debido a la naturaleza delicada del problema, es decir, los problemas que provienen de la naturaleza parabólica del punto fijo y la naturaleza exponencialmente pequeña de la separación, el control cuidadoso de los errores del "separatrix map" es primordial. Esto se logra empleando métodos geométricos, es decir, aprovechando al máximo las funciones generadoras que existen en virtud de la naturaleza simpléctica del sistema. Finalmente, motivados por el importante papel de la geometría simpléctica en los sistemas de mecánica celeste en mente, damos ejemplos de estructuras simplécticas degeneradas y singulares que ocurren en sistemas de mecánica celeste que no pueden equiparse con una forma simpléctica.Matemàtica aplicad

An Invitation to Singular Symplectic Geometry

In this paper we analyze in detail a collection of motivating examples to consider $b^m$-symplectic forms and folded-type symplectic structures. In particular, we provide models in Celestial Mechanics for every $b^m$-symplectic structure. At the end of the paper, we introduce the odd-dimensional analogue to $b$-symplectic manifolds: $b$-contact manifolds.Comment: 14 pages, 1 figur

Strain-induced structural instability in FeRh

We perform density functional calculations to investigate the structure of the inter-metallic alloy FeRh under epitaxial strain. Bulk FeRh exhibits a metamagnetic transition from a low-temperature antiferromagnetic (AFM) phase to a ferromagnetic (FM) phase at 350K, and its strain dependence is of interest for tuning the transition temperature to the room-temperature operating conditions of typical memory devices. We find an unusually strong dependence of the structural energetics on the choice of exchange-correlation functional, with the usual local density approximation (LDA) yielding the wrong ground-state structure, and generalized gradient (GGA) extensions being in better agreement with the bulk experimental structure. Using the GGA we show the existence of a metastable face-centered-cubic (fcc)-like AFM structure that is reached from the ground state body-centered-cubic (bcc) AFM structure by following the epitaxial Bain path. We predict that this metastable fcc-like structure has a significantly higher conductivity than the bcc AFM phase. We show that the behavior is well described using non-linear elasticity theory, which captures the softening and eventual sign change of the orthorhombic shear modulus under compressive strain, consistent with this structural instability. Finally, we predict the existence of an additional unit-cell-doubling lattice instability, which should be observable at low temperature.Comment: 10 pages, 7 figure

Cotangent models for group actions on bb-Poisson manifolds

In this article we give a normal form of a $b$-symplectic form in the neighborhood of a compact orbit of a Lie group action on a $b$-symplectic manifold. We establish cotangent models for Poisson actions on $b$-Poisson manifolds and a $b$-symplectic slice theorem. We examine interesting particular instances of group actions on $b$-symplectic manifolds preserving the Poisson structure. Also, we revise the notion of cotangent lift and twisted $b$-cotangent lift introduced in [KM] and provide a generalization of the twisted $b$-cotangent lift. We introduce the notion of $b$-Lie group and the associated $b$-symplectic structures in its $b$-cotangent bundle together with their reduction theory.Comment: 19 pages, 1 figur

Pátio, como modo de construir

Dissertação de mestrado integrado em Arquitectura, Universidade Lusíada de Lisboa, 2014Exame público realizado em 14 de Maio de 2014A presente dissertação pretende mostrar o pátio, como modo de construir. Tomando como exemplo diversas escalas a presente dissertação tem como objetivo mostrar este modo de construir tão transversal ao longo da história da arquitetura. Olhando através de várias experiências e de vários exemplos, deixamos o nosso olhar prender-se com três momentos distintos e apartir desses momentos damos início à construção do nosso imaginário do projeto

Cavity-enhanced absorption detection of H2S in the near-infrared using a gain-switched frequency comb laser

A custom-designed gain-switched frequency comb laser was passively coupled of to a medium-finesse cavity in the region between 6346 and 6354 cm−1 for the development of a prototype cavity enhanced absorption setup. The setup was applied to static gas detection of hydrogen sulfide at the parts per thousand level in a laboratory environment. A Fourier transform spectrometer was used for signal detection. The experimental performance of the setup was characterized in this proof-of-principle investigation; advantages, drawbacks and future scope of the approach are discussed in this article

Compact gain switched optical frequency comb generator for sensing applications

We present a novel InP photonically integrated optically injected device that is gain switched for the generation of an optical frequency comb. Using this technique, an optical frequency comb with a free spectral range of 6.25 GHz and nine spectral lines within a 3 dB spectral window is obtained. Such a device provides tunability of both the free spectral range and the centre emission wavelength, which facilitates the matching of the wavelength to the signature of a target gas. The stable spacing and high phase correlation between the comb lines confirms the potential of the device to be used in various applications such as spectroscopy, telecommunications and gas sensing

Increasing Smoking Cessation Counseling in Patients with Substance Use Disorders

The United States reports more opioid-related deaths than any other country across the globe, with over 90,000 lives lost in the year 2020. Patients undergoing MAT have exponentially higher rates of tobacco and nicotine use compared to the public, in fact, five times greater than the national average. At large, these patients commonly forego smoking cessation (SC) counseling or SC products, nor do they adhere to SC programs. The use of multifaceted interventions appears to be the most effective means of SC counseling techniques within the MAT population. Combination interventions, such as the use of nicotine replacement therapy (NRT) with motivational interviewing or behavioral reduction methods, not only improved rates of quit attempts but also produced long-term abstinence. The introduction of multiple components of SC techniques early in a patient's substance use recovery offer a patient the greatest benefits and chance for successful smoking abstinence. This quality improvement project aims to educate clinicians upon the importance of a multifaceted intervention aimed to engage patients in SC counseling while increasing readiness to quit smoking. After implementation of the educational sessions, the occurrence of SC counseling increased by 3% within this clinical site. This project shows that even greater efforts are required to encourage MAT patients and their clinicians to commit to SC efforts

New geometrical and dynamical techniques for problems in celestial mechanics

In this thesis, we study the application of symplectic geometry, regular and singular, to symplectic dynamical systems.We start with a motivating case: the relation between symplectic foliations and global transverse Poincaré sections, showing that meaningful dynamical information can be gleaned by simple observations on the geometry of the phase space - in this case, the existence of a symplectic foliation on a hypersurface of the phase space. We then go on study dynamical systems of particular importance in geometry - those given by a group action on a manifold. In particular, we consider a singular symplectic manifold (specifically, a manifold equipped with a symplectic form which blows up in a controlled manner on a hypersurface of that manifold, namely, a b-symplectic form) with a group action preserving the geometry and give a b-symplectic slice theorem which provides an equivariant normal form of the b-symplectic form in the neighbourhood of an orbit. Particular examples of b-symplectic group symmetries are then explored: those given by the cotangent lift of group translation on so-called b-Lie groups. The second part of this thesis focuses on symplectic and b-symplectic dynamical systems coming from celestial mechanics. In particular, the separatrix map of the stable and unstable manifolds of the fixed point at infinity of the planar circular restricted three-body problem is examined and an estimate of the width of the stochastic layer is given - that is the existence of a K.A.M. torus which acts as a boundary to bounded motions is proved. Due to the delicate nature of the problem - namely issues coming from the parabolic nature of the fixed point and exponentially small nature of the splitting, careful control of the errors of the separatrix map is paramount. This is achieved by employing geometric methods, namely, by taking full advantage of generating functions which exist by virtue of the symplectic nature of the system. Finally, motivated by the important role of symplectic geometry in the systems of celestial mechanics in mind, we give examples of degenerate and singular symplectic structures occurring in systems of celestial mechanics which cannot be equipped with a symplectic form.En esta tesis, estudiamos una aplicación de la geometría simpléctica, regular y singular, a sistemas dinámicos simplécticos. Comenzamos con un caso motivador: la relación entre foliaciones simplécticas y secciones transversales de Poincaré globales, que muestra que se puede obtener información significativa de la dinámica mediante simples observaciones sobre la geometría del espacio de fase, en este caso, la existencia de una foliación simpléctica en una hipersuperficie del espacio de fase. Luego continuamos estudiando sistemas dinámicos de particular importancia en geometría, dados por una acción de grupo sobre una variedad. En particular, consideramos una variedad simpléctica singular (específicamente, una variedad equipada con una forma simpléctica que explota de manera controlada en una hipersuperficie de esa variedad, es decir, una forma $b$-simpléctica ) con una acción de grupo que preserva la geometría y damos un teorema de rebanada $b$-simpléctico que proporciona una forma normal equivariante de la forma b-simpléctica en una vecindad de una órbita. Luego se exploran ejemplos particulares de simetrías de grupo b-simplécticas: las dadas por el levantamiento cotangente de la translación (grupal) en los llamados b-grupos de Lie. La segunda parte de esta tesis se enfoca en sistemas dinámicos simplécticos y b-simplécticos provenientes de la mecánica celeste. En particular, se examina el "separatrix map" de variedades estables e inestables del punto fijo al infinito del problema de los tres cuerpos restringido circular plano, y se da una estimación del ancho de la capa estocástica, es decir, la existencia de un toro K.A.M. que actúa como frontera para movimientos acotados. Debido a la naturaleza delicada del problema, es decir, los problemas que provienen de la naturaleza parabólica del punto fijo y la naturaleza exponencialmente pequeña de la separación, el control cuidadoso de los errores del "separatrix map" es primordial. Esto se logra empleando métodos geométricos, es decir, aprovechando al máximo las funciones generadoras que existen en virtud de la naturaleza simpléctica del sistema. Finalmente, motivados por el importante papel de la geometría simpléctica en los sistemas de mecánica celeste en mente, damos ejemplos de estructuras simplécticas degeneradas y singulares que ocurren en sistemas de mecánica celeste que no pueden equiparse con una forma simpléctica.Postprint (published version