Some notes to extend the study on random non-autonomous second order linear differential equations appearing in Mathematical Modeling

The objective of this paper is to complete certain issues from our recent contribution [J. Calatayud, J.-C. Cort\'es, M. Jornet, L. Villafuerte, Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties, Advances in Difference Equations, 2018:392, 1--29 (2018)]. We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via an habitual Lipschitz condition that extends the classical Picard Theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, being the latter a reformulation of our random Fr\"obenius method.Comment: 15 pages, 0 figures, 2 table

Analysis of the random heat equation via approximate density functions

[EN] In this paper we study the randomized heat equation with homogeneous boundary conditions. The diffusion coefficient is assumed to be a random variable and the initial condition is treated as a stochastic process. The solution of this randomized partial differential equation problem is a stochastic process, which is given by a random series obtained via the classical method of separation of variables. Any stochastic process is determined by its finite-dimensional joint distributions. In this paper, the goal is to obtain approximations to the probability density function of the solution (the first finite-dimensional distributions) under mild conditions. Since the solution is expressed as a random series, we perform approximations to its probability density function. Several illustrative examples are shown.This work has been supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P.Calatayud, J.; Cortés, J. (2021). Analysis of the random heat equation via approximate density functions. Romanian Reports in Physics. 73(2):1-10. http://hdl.handle.net/10251/181144S11073

Cognitive differences in the older adults living in the general community: gender and mental occupational state study

Older adults are particularly vulnerable to cognitive impairment with age, and gender differences are remarkable. However, there is very little evidence to identify both baseline cognitive and occupational gender differences prior to older adults’ retirement to design more efficient personalized cognitive interventions. This descriptive observational study examined gender differences in initial cognitive performance in 367 older adults with subjective memory complaints from a primary healthcare center in Zaragoza (Spain). To evaluate initial cognitive performance, the Spanish version of the Mini-Mental State Examination (MEC-35) and the set test were used to measure verbal fluency. Sociodemographic and clinical characteristics were evaluated, and cognitive and occupational differences were analyzed per gender. Men had higher educational and occupational levels, were older and more of them were married (p < 0.001) than women. Regarding cardiovascular risk factors, diabetes and cerebrovascular accidents were more frequent in women, while hypercholesterolemia and obesity were more frequent in men (p < 0.001). High blood pressure was more frequent in women, but not significantly so (p = 0.639). Global cognition was higher in men (p < 0.001) for attention, calculation, and language (p < 0.001). Verbal fluency was higher in women, but the difference was not statistically significant (p = 0.105). These results could be gen-eralized to other health centers in the province and other Spanish autonomous communities as their sociodemographic variables are similar. Individualized interventions that adapt to gender, cognitive and initial occupational performance should be developed and adapted to elderly populations living in the general community to maintain their cognitive capacity and prevent their cognitive impairment and the social health costs this would imply

Application of modulated chlorophyll fluorescence and modulated chlorophyll fluorescence imaging to study the environmental stress effect

Chlorophyll (Chl) a fluorescence is a widely used tool to monitor the photosynthetic process in plants subjected to environmental stresses. this review reports the theoretical bases of Chl fluorescence, and the significance of the most important Chl fluorescence parameters. it also reports how these parameters can be utilised to estimate changes in photosystem (Ps) ii photochemistry, linear electron flux and dissipation mechanisms. the relation between actual Psii photochemistry and Co2 assimilation is discussed, as is the role of photochemical and non-photochemical quenching in inducing changes in Psii activity. the application of Chl fluorescence imaging to study heterogeneity on leaf lamina is also considered. this review summarises only some of the results obtained by this methodology to study the effects of different environmental stresses, namely water availability, nutrients, pollutants, temperature and salinity

Random differential equations with discrete delay

[EN] In this article, we study random differential equations with discrete delay with initial condition The uncertainty in the problem is reflected via the outcome omega. The initial condition g(t) is a stochastic process. The term x(t) is a stochastic process that solves the random differential equation with delay in a probabilistic sense. In our case, we use the random calculus approach. We extend the classical Picard theorem for deterministic ordinary differential equations to calculus for random differential equations with delay, via Banach fixed-point theorem. We also relate solutions with sample-path solutions. Finally, we utilize the theoretical ideas to solve the random autonomous linear differential equation with discrete delay.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017 89664 PCalatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Random differential equations with discrete delay. Stochastic Analysis and Applications. 37(5):699-707. https://doi.org/10.1080/07362994.2019.1608833S699707375Fridman, E., & Shaikhet, L. (2017). Stabilization by using artificial delays: An LMI approach. Automatica, 81, 429-437. doi:10.1016/j.automatica.2017.04.015Shaikhet, L., & Korobeinikov, A. (2015). Stability of a stochastic model for HIV-1 dynamics within a host. Applicable Analysis, 95(6), 1228-1238. doi:10.1080/00036811.2015.1058363Caraballo, T., Colucci, R., & Guerrini, L. (2018). On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 17(6), 2703-2727. doi:10.3934/cpaa.2018128Caraballo, T., J. Garrido-Atienza, M., Schmalfuss, B., & Valero, J. (2017). Attractors for a random evolution equation with infinite memory: Theoretical results. Discrete & Continuous Dynamical Systems - B, 22(5), 1779-1800. doi:10.3934/dcdsb.2017106Krapivsky, P. L., Luck, J. M., & Mallick, K. (2011). On stochastic differential equations with random delay. Journal of Statistical Mechanics: Theory and Experiment, 2011(10), P10008. doi:10.1088/1742-5468/2011/10/p10008Liu, S., Debbouche, A., & Wang, J. (2017). On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. Journal of Computational and Applied Mathematics, 312, 47-57. doi:10.1016/j.cam.2015.10.028Dorini, F. A., Cecconello, M. S., & Dorini, L. B. (2016). On the logistic equation subject to uncertainties in the environmental carrying capacity and initial population density. Communications in Nonlinear Science and Numerical Simulation, 33, 160-173. doi:10.1016/j.cnsns.2015.09.009Slama, H., El-Bedwhey, N. A., El-Depsy, A., & Selim, M. M. (2017). Solution of the finite Milne problem in stochastic media with RVT Technique. The European Physical Journal Plus, 132(12). doi:10.1140/epjp/i2017-11763-6Nouri, K., Ranjbar, H., & Torkzadeh, L. (2019). Modified stochastic theta methods by ODEs solvers for stochastic differential equations. Communications in Nonlinear Science and Numerical Simulation, 68, 336-346. doi:10.1016/j.cnsns.2018.08.013Lupulescu, V., O’Regan, D., & ur Rahman, G. (2014). Existence results for random fractional differential equations. Opuscula Mathematica, 34(4), 813. doi:10.7494/opmath.2014.34.4.813Strand, J. . (1970). Random ordinary differential equations. Journal of Differential Equations, 7(3), 538-553. doi:10.1016/0022-0396(70)90100-2Villafuerte, L., Braumann, C. A., Cortés, J.-C., & Jódar, L. (2010). Random differential operational calculus: Theory and applications. Computers & Mathematics with Applications, 59(1), 115-125. doi:10.1016/j.camwa.2009.08.061Granas, A., & Dugundji, J. (2003). Fixed Point Theory. Springer Monographs in Mathematics. doi:10.1007/978-0-387-21593-

Mathematical methods for the randomized non-autonomous Bertalanffy model

[EN] In this article we analyze the randomized non-autonomous Bertalanffy model x' (t, omega) = a(t, omega)x(t, omega) b(t, omega)x(t, omega)(2/3), x(t(0), omega) = x(0)(omega), where a(t, omega) and b(t, omega) are stochastic processes and x(0)(omega) is a random variable, all of them defined in an underlying complete probability space. Under certain assumptions on a, b and x(0), we obtain a solution stochastic process, x(t, omega), both in the sample path and in the mean square senses. By using the random variable transformation technique and Karhunen-Loeve expansions, we construct a sequence of probability density functions that under certain conditions converge pointwise or uniformly to the density function of x(t, omega), f (t) (x). This permits approximating the expectation and the variance of x(t, omega). At the end, numerical experiments are carried out to put in practice our theoretical findings.This work was supported by the Spanish Ministerio de Economia, Industria y Competitividad (MINECO), by the Agencia Estatal de Investigacion (AEI) and Fondo Europeo de Desarrollo Regional (FEDER UE) grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud, J.; Caraballo, T.; Cortés, J.; Jornet, M. (2020). Mathematical methods for the randomized non-autonomous Bertalanffy model. Electronic Journal of Differential Equations. 2020:1-19. http://hdl.handle.net/10251/161056S119202