Production and characterization of the Cr_35Fe_35V_16.5Mo_6Ti_7.5 high entropy alloy

The microstructure, thermal stability, and mechanical properties of a novel Cr_35Fe_35V_16.5Mo_6Ti_7.5 high-entropy alloy were studied. The mechanical properties were mapped by nanoindentation, and the results correlated with the microstructure and the Vickers microhardness measurements. The alloy was produced by arc melting in a low pressure He atmosphere. Thermal treatments were performed to study the thermal stability of the alloy. The as-cast microstructure of the alloy exhibited a body-centered cubic phase with morphology of dendrites, outlined by a very thin interdendritic phase with a crystallographic structure compatible with Fe_2Ti. The presence of the intermetallic particles was predicted by a free-energy based model, in contrast with the single solid solution alloy predicted by a parameter-based model. The volume fraction of the dendrites in the alloy is -94 % after arc melting. A small fraction of sparse Ti-rich particles, -0.4 vol%, was observed. The thermal treatments produced an increase of the population of Ti-rich particles, the formation of a sigma-phase and nucleation of precipitates enriched with Fe and Ti into the previous dendrites. The material in as-cast condition exhibited a microhardness value of 6.2 +-0.3 GPa, while the alloy aged at 960 ºC resulted in 7.1 +-0.4 GPa. Nanoindentations maps showed an excellent correlation with the microstructure, and their statistical analyses yielded a nanohardness mean value of 8.2 +-0.4 GPa in the dendritic BCC regions of the as-cast and thermal treated samples and 14.1 +-0.6 GPa for the sigma-phase. The onset of the plastic behavior has been studied by analyzing the pop-in phenomenon observed in the nanoindentation loading curves. For the as-cast alloy, this analysis showed that the elastic-to-plastic transition seems to be triggered by dislocation nucleation. The alloy has a low thermal diffusivity in the measured temperature range that increases on increasing temperature

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

Estimating health system opportunity costs: the role of non-linearities and inefficiency

BACKGROUND: Empirical estimates of health system opportunity costs have been suggested as a basis for the cost-effectiveness threshold to use in Health Technology Assessment. Econometric methods have been used to estimate these in several countries based on data on spending and mortality. This study examines empirical evidence on four issues: non-linearity of the relationship between spending and mortality; the inclusion of outcomes other than mortality; variation in the efficiency with which expenditures generate health outcomes; and the relationship among efficiency, mortality rates and outcome elasticities. METHODS: Quantile Regression is used to examine non-linearities in the relationship between mortality and health expenditures along the mortality distribution. Data Envelopment Analysis extends the approach, using multiple measures of health outcomes to measure efficiency. These are applied to health expenditure data from 151 geographical units (Primary Care Trusts) of the National Health Service in England, across eight different clinical areas (Programme Budget Categories), for 3 fiscal years from 2010/11 to 2012/13. RESULTS: The results suggest differences in efficiency levels across geographical units and clinical areas as to how health resources generate outcomes, which indicates the capacity to adjust to a decrease in health expenditure without affecting health outcomes. Moreover, efficient units have lower absolute levels of mortality elasticity to health expenditure than inefficient ones. CONCLUSIONS: The policy of adopting thresholds based on estimates of a single system-wide cost-effectiveness threshold assumes a relationship between expenditure and health outcomes that generates an opportunity cost estimate which applies to the whole system. Our evidence of variations in that relationship and therefore in opportunity costs suggests that adopting a single threshold may exacerbate the efficiency and equity concerns that such thresholds are designed to counter. In most health care systems, many decisions about provision are not made centrally. Our analytical approach to understanding variability in opportunity cost can help policy makers target efficiency improvements and set realistic targets for local and clinical area health improvements from increased expenditure. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1186/s12962-022-00391-y