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

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

On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached

Rational KdV potentials and differential galois theory

In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schrödinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems associated to the Schrödinger equation. Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformation

Estudio experimental del relleno de cavidades óseas con hidroxiapatita asociada a colágeno

La cirugía ortopédica y maxilo-facial necesita realizar en muchas ocasiones resecciones masivas de tejido óseo. Esto ha hecho que se hayan propuesta la utilización de materiales inertes como sustitutos óseos, gracias a la habilidad que tienen de permitir la regeneración del hueso, tanto en el campo de la medicina como de la odontología. Nuestro trabajo tiene por objetivo estudiar un biomaterial compuesto de hidroxiapatita asociado a colágeno como material de sustitución ósea (Collapat de OSTEO AG) en forma de esponjas, con un peso de 500 mg y de un tamaño de 35x30x6 mm. Se han intervenido 50 animales de experimentación (conejo Albino de Nueva Zelanda), divididos por 16 animales. Los animales de este grupo fueron intervenidos quirúrgicamente realizándose una cavidad a nivel metafisodiafisario en el fémur y otra en la tibia, pero en ninguna de las cavidades se realizaron implantes. Los animales del grupo II o grupo estudio estaban comprendidos por 34 animales, a los que se les realizó la misma intervención, pero se le implantó una esponja de hidroxiapatita-colágeno de las dimensiones antes descritas, en cada una de las cavidades. El estudio ha demostrado una regeneración ósea de la cavidad rellenada a expensas de la rápida reabsorbción del colágeno, y de una lenta reabsorbción de la hidroxiapatita