A mean square chain rule and its application in solving the random Chebyshev differential equation

[EN] In this paper a new version of the chain rule for calculat- ing the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the cor- responding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included.This work was completed with the support of our TEX-pert.Cortés, J.; Villafuerte, L.; Burgos-Simon, C. (2017). A mean square chain rule and its application in solving the random Chebyshev differential equation. Mediterranean Journal of Mathematics. 14(1):14-35. https://doi.org/10.1007/s00009-017-0853-6S1435141Calbo, G., Cortés, J.C., Jódar, L., Villafuerte, L.: Analytic stochastic process solutions of second-order random differential equations. Appl. Math. Lett. 23(12), 1421–1424 (2010). doi: 10.1016/j.aml.2010.07.011El-Tawil, M.A., El-Sohaly, M.: Mean square numerical methods for initial value random differential equations. Open J. Discret. Math. 1(1), 164–171 (2011). doi: 10.4236/ojdm.2011.12009Khodabin, M., Maleknejad, K., Rostami, K., Nouri, M.: Numerical solution of stochastic differential equations by second order Runge Kutta methods. Math. Comp. Model. 59(9–10), 1910–1920 (2010). doi: 10.1016/j.mcm.2011.01.018Santos, L.T., Dorini, F.A., Cunha, M.C.C.: The probability density function to the random linear transport equation. Appl. Math. Comput. 216(5), 1524–1530 (2010). doi: 10.1016/j.amc.2010.03.001González Parra, G., Chen-Charpentier, B.M., Arenas, A.J.: Polynomial Chaos for random fractional order differential equations. Appl. Math. Comput. 226(1), 123–130 (2014). doi: 10.1016/j.amc.2013.10.51El-Beltagy, M.A., El-Tawil, M.A.: Toward a solution of a class of non-linear stochastic perturbed PDEs using automated WHEP algorithm. Appl. Math. Model. 37(12–13), 7174–7192 (2013). doi: 10.1016/j.apm.2013.01.038Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterran. J. Math. 12(3), 1123–1140 (2015). doi: 10.1007/s00009-014-0452-8Villafuerte, L., Braumann, C.A., Cortés, J.C., Jódar, L.: Random differential operational calculus: theory and applications. Comp. Math. Appl. 59(1), 115–125 (2010). doi: 10.1016/j.camwa.2009.08.061Øksendal, B.: Stochastic differential equations: an introduction with applications, 6th edn. Springer, Berlin (2007)Soong, T.T.: Random differential equations in science and engineering. Academic Press, New York (1973)Wong, B., Hajek, B.: Stochastic processes in engineering systems. Springer Verlag, New York (1985)Arnold, L.: Stochastic differential equations. Theory and applications. John Wiley, New York (1974)Cortés, J.C., Jódar, L., Camacho, J., Villafuerte, L.: Random Airy type differential equations: mean square exact and numerical solutions. Comput. Math. Appl. 60(5), 1237–1244 (2010). doi: 10.1016/j.camwa.2010.05.046Calbo, G., Cortés, J.C., Jódar, L.: Random Hermite differential equations: mean square power series solutions and statistical properties. Appl. Math. Comp. 218(7), 3654–3666 (2011). doi: 10.1016/j.amc.2011.09.008Calbo, G., Cortés, J.C., Jódar, L., Villafuerte, L.: Solving the random Legendre differential equation: Mean square power series solution and its statistical functions. Comp. Math. Appl. 61(9), 2782–2792 (2010). doi: 10.1016/j.camwa.2011.03.045Cortés, J.C., Jódar, L., Company, R., Villafuerte, L.: Laguerre random polynomials: definition, differential and statistical properties. Utilit. Math. 98, 283–293 (2015)Cortés, J.C., Jódar, L., Villafuerte, L.: Mean square solution of Bessel differential equation with uncertainties. J. Comp. Appl. Math. 309, 383–395 (2017). doi: 10.1016/j.cam.2016.01.034Golmankhaneh, A.K., Porghoveh, N.A., Baleanu, D.: Mean square solutions of second-order random differential equations by using homotopy analysis method. Romanian Reports Physics 65(2), 1237–1244 (2013)Khalaf, S.L.: Mean square solutions of second-order random differential equations by using homotopy perturbation method. Int. Math. Forum 6(48), 2361–2370 (2011)Khudair, A.R., Ameen, A.A., Khalaf, S.L.: Mean square solutions of second-order random differential equations by using Adomian decomposition method. Appl. Math. Sci. 5(49), 2521–2535 (2011)Agarwal, R.P., O’Regan, D.: Ordinary and partial differential equations. Springer, New York (2009

Minimising Mortality in Endangered Raptors Due to Power Lines: The Importance of Spatial Aggregation to Optimize the Application of Mitigation Measures

Electrocution by power lines is one of the main causes of non-natural mortality in birds of prey. In an area in central Spain, we surveyed 6304 pylons from 333 power lines to determine electrocution rates, environmental and design factors that may influence electrocution and the efficacy of mitigation measures used to minimise electrocution cases. A total of 952 electrocuted raptors, representing 14 different species, were observed. Electrocuted raptors were concentrated in certain areas and the environmental factors associated with increased electrocution events were: greater numbers of prey animals; greater vegetation cover; and shorter distance to roads. The structural elements associated with electrocutions were shorter strings of insulators, one or more phases over the crossarm, cross-shaped design and pylon function. Of the 952 carcasses found, 148 were eagles, including golden eagle (Aquila chrysaetos), Spanish imperial eagle (Aquila adalberti) and Bonelli's eagle (Aquila fasciata). Electrocuted eagles were clustered in smaller areas than other electrocuted raptors. The factors associated with increased eagle electrocution events were: pylons function, shorter strings of insulators, higher slopes surrounding the pylon, and more numerous potential prey animals. Pylons with increased string of insulators had lower raptor electrocution rates than unimproved pylons, although this technique was unsuccessful for eagles. Pylons with cable insulation showed higher electrocution rates than unimproved pylons, both for raptors and eagles, despite this is the most widely used and recommended mitigation measure in several countries. To optimize the application of mitigation measures, our results recommend the substitution of pin-type insulators to suspended ones and elongating the strings of insulators

The genetics of addiction—a translational perspective

Addictions are serious and common psychiatric disorders, and are among the leading contributors to preventable death. This selective review outlines and highlights the need for a multi-method translational approach to genetic studies of these important conditions, including both licit (alcohol, nicotine) and illicit (cannabis, cocaine, opiates) drug addictions and the behavioral addiction of disordered gambling. First, we review existing knowledge from twin studies that indicates both the substantial heritability of substance-specific addictions and the genetic overlap across addiction to different substances. Next, we discuss the limited number of candidate genes which have shown consistent replication, and the implications of emerging genomewide association findings for the genetic architecture of addictions. Finally, we review the utility of extensions to existing methods such as novel phenotyping, including the use of endophenotypes, biomarkers and neuroimaging outcomes; emerging methods for identifying alternative sources of genetic variation and accompanying statistical methodologies to interpret them; the role of gene-environment interplay; and importantly, the potential role of genetic variation in suggesting new alternatives for treatment of addictions