Examinando por Autor "Morales-Ruiz, Juan J."
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Ítem Differential Galois theory and non-integrability of planar polynomial vector fields(Elsevier, 2018-06) Acosta-Humánez, Primitivo B.; Lázaro, J. Tomás; Morales-Ruiz, Juan J.; Pantazi, CharaWe study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrabilitywith some higher transcendent functions, like the error function.Ítem Integrability of stochastic birth-death processes via differential galois theory(EDP Sciences, 2020) Acosta-Humánez, Primitivo B.; Capitán, José A.; Morales-Ruiz, Juan J.Stochastic birth-death processes are described as continuous-time Markov processes in models of population dynamics. A system of in nite, coupled ordinary diferential 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 diferential equation. In this contribution we analyze the integrability of two types of stochastic birth-death processes (with polynomial birth and death rates) using standard diferential 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 case in which rates are linear functions of the number of individuals.