Integrability of Stochastic Birth-Death processes via Differential Galois Theory

Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of infinite, coupled ordinary differential equations (the so-called master equation) describes the time-dependence of the probability of each system state. Using a generating function, the master equation can be transformed into a partial differential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard differential Galois theory. We discuss the integrability of the PDE via a Laplace transform acting over the temporal variable. We show that the PDE is not integrable except for the (trivial) case in which rates are linear functions of the number of individuals

Two-dimensional approach to relativistic positioning systems

A relativistic positioning system is a physical realization of a coordinate system consisting in four clocks in arbitrary motion broadcasting their proper times. The basic elements of the relativistic positioning systems are presented in the two-dimensional case. This simplified approach allows to explain and to analyze the properties and interest of these new systems. The positioning system defined by geodesic emitters in flat metric is developed in detail. The information that the data generated by a relativistic positioning system give on the space-time metric interval is analyzed, and the interest of these results in gravimetry is pointed out.Comment: 11 pages, 5 figures. v2: a brief description of the principal bibliography has been adde

A note on the geodesic deviation equation for null geodesics in the Schwarzschild black-hole

We use the Hamiltonian formulation of the geodesic equation in the Schwarzschild space-time so as to get the variational equation as the counterpart of the Jacobi equation in this approach. In this context we are able to apply the Morales-Ramis theorem to link the integrability of the geodesic equation to the integrability, in the sense of differential Galois theory, of the variational equation. This link is strong enough to hold even on geodesics for which the usual conserved quantities fail to be independent, as is the case of circular geodesics. We show explicitly the particular cases of some null geodesics and their variational equations.Comment: 12 page

Galoisian Approach to integrability of Schr\"odinger Equation

In this paper, we examine the non-relativistic stationary Schr\"odinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second order ordinary linear differential operators, so as to achieve rational function coefficients ("algebrization"), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schr\"odinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.Comment: 62 page

Práctica de diseño hardware/software de un robot móvil con interfaces inalámbricas

En el presente artículo se describe una práctica de laboratorio multitarea en el ámbito de las asignaturas de sistemas empotrados en los grados de Ingeniería Informática, mediante una metodología de gestión de proyectos basada en Kanban. La práctica abarca diferentes familias de microcontroladores de distintos niveles de dificultad de programación, lectura de diferentes tipos de sensores con distintas interfaces, comunicación inalámbrica y control de motores. Esta práctica se enfoca como la elaboración de un proyecto en el que los alumnos han de ir realizando mediante tareas que inicialmente se planifican utilizando la metodología Kanban. En concreto, el desarrollo de la práctica se basa en la elaboración de un robot móvil controlado remotamente y de forma inalámbrica. El sistema de divide en tres partes: el dispositivo de control que cuenta con un microcontrolador tipo Arduino y dos joysticks analógicos como interfaz de usuario, el robot móvil que utiliza un microcontrolador STM32 con un RTOS (Real Time Operating System) con el que se realiza la lectura de los diferentes sensores que irán embarcados en el robot además de manejar el controlador de motores para un motor DC para la velocidad y un servo para el control de la dirección. Para la comunicación inalámbrica se utilizan módulos de radio de 2.4GHz de la familia XBee Pro Serie Z2B. Por último, se diseñará una aplicación software de escritorio bajo un sistema operativo Windows escrita en lenguaje C# utilizando .NET Framework y WPF (Windows Presentation Foundation), que mostrará la información que el robot envía de cada uno de sus sensores. El PC donde está alojada la aplicación tiene conectado un módulo XBee, anteriormente mencionado, con el que se comunica mediante una conexión serie virtual (VCP). Para implementar la metodología Kanban se hará uso de una herramienta online y gratuita llamada Trello que permite la creación de diferentes tableros en el que ir añadiendo tareas (mediante tarjetas) e irlas moviendo entre las diferentes columnas según el estado de ésta. A cada tarea se le puede añadir uno o más participantes además de ponerle una fecha de vencimiento entre otras opciones. En el desarrollo de este tipo de prácticas se añade la dificultad del manejo de diferentes entornos de desarrollo, uno por cada tipo de microcontrolador y el de la aplicación software. Esta práctica se ha dividido en varias sesiones y ha presentado un gran atractivo para el alumnado ya que se consigue un sistema funcional y muy ampliable al final de estas.This paper presents a laboratory session of embedded systems imparted in the Computer Science degree using Kanban, a project management methodology. In the laboratory session different microcontroller families are used for reading several sensor types, wireless communications and motor control. This session is focused like a project in which the students have to complete the task previously described using Kanban. The project consist on implementing a mobile robot that is handled using a wireless controller. The system is divided in three parts: the controller device that is designed using an Arduino microcontroller to read two analogical joysticks used by the user, the mobile robot that uses a STM32 microcontroller with a RTOS (Real Time Operating System) to read the sensors attached to the robot and to handle the motor controller for a DC motor to control the velocity and, finally, a servo motor to change the robot direction. Some 2.4GHz radio modules of the XBee Pro Serie Z2B are used to implement the wireless communication. Finally a C# WPF Windows application is implemented using .NET framework, which collects the information from on-board sensors. An XBee module is plugged in the computer where the application runs using a virtual communication port (VCP). To plan the project under the Kanban methodology, an online free tool called Trello is used. Trello lets the user create different panels in which cards can be added and moved between different columns that denote the state of each card. Cards allow to add several participants and a due date. In this laboratory session the students have to learn several development environments which presents an extra difficulty. The laboratory session has been divided in several practical sessions and the students have been very motivated during every of them because at the end they obtain a functional robot which can be extended with new sensors

Temperature and pressure constraints near the freezing point

The isothermal-isobaric ensemble molecular-dynamics method MD(T,p,N) proposed by Nosé and Hoover is used to study the fluctuations in a two-dimensional Lennard-Jones fluid, close to the freezing point. The T and p constraints in this method do not affect the dynamical behavior of the system, since spontaneous fluctuations in the density allow the system to freeze and melt just as do the T and p fluctuations in the microcanonical ensemble MD(E,V,N) close to the melting zone