Understanding the Hamiltonian Function through the Geometry of Partial Legendre Transforms

The relationship between the Hamiltonian and Lagrangean functions in analytical mechanics is a type of duality. The two functions, while distinct, are both descriptive functions encoding the behavior of the same dynamical system. One difference is that the Lagrangean naturally appears as one investigates the fundamental equation of classical dynamics. It is not that way for the Hamiltonian. The Hamiltonian comes after Lagrange's equations have been fully formed, most commonly through a Legendre transform of the Lagrangean function. We revisit the Legendre transform approach and offer a more refined geometrical interpretation than what is commonly shown

Investigation of vertical cavity surface emitting laser dynamics for neuromorphic photonic systems

We report an approach based upon vertical cavity surface emitting lasers (VCSELs) to reproduce optically different behaviors exhibited by biological neurons but on a much faster timescale. The technique proposed is based on the polarization switching and nonlinear dynamics induced in a single VCSEL under polarized optical injection. The particular attributes of VCSELs and the simple experimental configuration used in this work offer prospects of fast, reconfigurable processing elements with excellent fan-out and scaling potentials for use in future computational paradigms and artificial neural networks. © 2012 American Institute of Physics

Relación de la escala de intensidad de Mercalli y la información instrumental como una tarea de clasificación de patrones

A pesar de los progresos ocurridos en la instrumentación sísmica, la valoración de vulnerabilidad sísmica y el daño con índices cualitativos, tal como los proporcionados por Intensidad de Mercalli Modificada (IMM), siguen siendo altamente favorables y útiles para los propósitos prácticos. Para vincular las medidas cualitativas de acción del terremoto y sus efectos, es habitualmente aplicada la regresión estadística. En este artículo, se adopta un planteamiento diferente, el cual consiste en expresar la Intensidad de Mercalli, como una clase en vez de un valor numérico. Una herramienta de clasificación estadística moderna, conocida como máquina de vectores de soporte, se usa para clasificar la información instrumental con el fin de evaluar la intensidad de Mercalli correspondiente. Se muestra que el método da resultados satisfactorios con respecto a las altas incertidumbres y a la medida del daño sísmico cualitativo

Breakdown of Hydrodynamics in a Simple One-Dimensional Fluid

We investigate the behavior of a one-dimensional diatomic fluid under a shock wave excitation. We find that the properties of the resulting shock wave are in striking contrast with those predicted by hydrodynamic and kinetic approaches, e.g., the hydrodynamic profiles relax algebraically toward their equilibrium values. Deviations from local thermodynamic equilibrium are persistent, decaying as a power law of the distance to the shock layer. Non-equipartition is observed infinitely far from the shock wave, and the velocity-distribution moments exhibit multiscaling. These results question the validity of simple hydrodynamic theories to understand collective behavior in 1d fluids.Comment: 4 pages, 5 figure

Pattern Recognition for a Flight Dynamics Monte Carlo Simulation

The design, analysis, and verification and validation of a spacecraft relies heavily on Monte Carlo simulations. Modern computational techniques are able to generate large amounts of Monte Carlo data but flight dynamics engineers lack the time and resources to analyze it all. The growing amounts of data combined with the diminished available time of engineers motivates the need to automate the analysis process. Pattern recognition algorithms are an innovative way of analyzing flight dynamics data efficiently. They can search large data sets for specific patterns and highlight critical variables so analysts can focus their analysis efforts. This work combines a few tractable pattern recognition algorithms with basic flight dynamics concepts to build a practical analysis tool for Monte Carlo simulations. Current results show that this tool can quickly and automatically identify individual design parameters, and most importantly, specific combinations of parameters that should be avoided in order to prevent specific system failures. The current version uses a kernel density estimation algorithm and a sequential feature selection algorithm combined with a k-nearest neighbor classifier to find and rank important design parameters. This provides an increased level of confidence in the analysis and saves a significant amount of time