R^2 Corrections and Non-perturbative Dualities of N=4 String ground states

We compute and analyse a variety of four-derivative gravitational terms in the effective action of six- and four-dimensional type II string ground states with N=4 supersymmetry. In six dimensions, we compute the relevant perturbative corrections for the type II string compactified on K3. In four dimensions we do analogous computations for several models with (4,0) and (2,2) supersymmetry. Such ground states are related by heterotic-type II duality or type II-type II U-duality. Perturbative computations in one member of a dual pair give a non-perturbative result in the other member. In particular, the exact CP-even R^2 coupling on the (2,2) side reproduces the tree-level term plus NS 5-brane instanton contributions on the (4,0) side. On the other hand, the exact CP-odd coupling yields the one-loop axionic interaction a.R\wedge R together with a similar instanton sum. In a subset of models, the expected breaking of the SL(2,Z)_S S-duality symmetry to a \Gamma(2)_S subgroup is observed on the non-perturbative thresholds. Moreover, we present a duality chain that provides evidence for the existence of heterotic N=4 models in which N=8 supersymmetry appears at strong coupling.Comment: Latex2e, 51 pages, 1 figur

Dispatcher3 – Machine learning to support flight planning processes

This poster will present the final results of the Clean Sky 2 project Dispatcher3. Dispatcher3 focuses on the use of machine learning techniques to support flight operations prior departure with holding predictions, runway at arrival estimation and fuel deviations pre-departure to support the flight crew, and ATFM and reactionary delays on D-1 to support the duty manage

Lp-calculus approach to the random autonomous linear differential equation with discrete delay

[EN] In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay , with initial condition x(t)=g(t), -t0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. Using Lp-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an Lp-solution too. An analysis of Lp-convergence when the delay tends to 0 is also performed in detail.This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Lp-calculus approach to the random autonomous linear differential equation with discrete delay. Mediterranean Journal of Mathematics. 16(4):1-17. https://doi.org/10.1007/s00009-019-1370-6S117164Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics. Springer, New York (2011)Driver, Y.: Ordinary and Delay Differential Equations. Applied Mathematical Science Series. Springer, New York (1977)Kuang, Y.: Delay Differential Equations: with Applications in Population Dynamics. Academic Press, Cambridge (2012)Bocharov, G.A., Rihan, F.A.: Numerical modelling in biosciences using delay differential equations. J. Comput. Appl. Math. 125, 183–199 (2000). https://doi.org/10.1016/S0377-0427(00)00468-4Jackson, M., Chen-Charpentier, B.M.: Modeling plant virus propagation with delays. J. Comput. Appl. Math. 309, 611–621 (2017). https://doi.org/10.1016/j.cam.2016.04.024Chen-Charpentier, B.M., Diakite, I.: A mathematical model of bone remodeling with delays. J. Comput. Appl. Math. 291, 76–84 (2016). https://doi.org/10.1016/j.cam.2017.01.005Erneux, T.: Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences Series. Springer, New York (2009)Kyrychko, Y.N., Hogan, S.J.: On the Use of delay equations in engineering applications. J. Vib. Control 16(7–8), 943–960 (2017). https://doi.org/10.1177/1077546309341100Matsumoto, A., Szidarovszky, F.: Delay Differential Nonlinear Economic Models (in Nonlinear Dynamics in Economics, Finance and the Social Sciences), 195–214. Springer-Verlag, Berlin Heidelberg (2010)Harding, L., Neamtu, M.: A dynamic model of unemployment with migration and delayed policy intervention. Comput. Econ. 51(3), 427–462 (2018). https://doi.org/10.1007/s10614-016-9610-3Oksendal, B.: Stochastic Differential Equations. Springer, New York (1998)Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, New York (2013)Hartung, F., Pituk, M.: Recent Advances in Delay Differential and Differences Equations. Springer-Verlag, Berlin Heidelberg (2014)Shaikhet, L.: Stability of equilibrium states of a nonlinear delay differential equation with stochastic perturbations. Int. J. Robust Nonlinear Control 27(6), 915–924 (2016). https://doi.org/10.1002/rnc.3605Shaikhet, L.: About some asymptotic properties of solution of stochastic delay differential equation with a logarithmic nonlinearity. Funct. Differ. Equ. 4(1–2), 57–67 (2017)Fridman, E., Shaikhet, L.: Delay-induced stability of vector second-order systems via simple Lyapunov functionals. Automatica 74, 288–296 (2016). https://doi.org/10.1016/j.automatica.2016.07.034Benhadri, M., Zeghdoudi, H.: Mean square asymptotic stability in nonlinear stochastic neutral Volterra-Levin equations with Poisson jumps and variable delays. Functiones et Approximatio Commentarii Mathematici 58(2), 157–176 (2018). https://doi.org/10.7169/facm/1657Nouri, K., Ranjbar, H.: Improved Euler-Maruyama method for numerical solution of the Itô stochastic differential systems by composite previous-current-step idea. Mediterr. J. Math. 15, 140 (2018). https://doi.org/10.1007/s00009-018-1187-8Santonja, F., Shaikhet, L.: Probabilistic stability analysis of social obesity epidemic by a delayed stochastic model. Nonlinear Anal. Real World Appl. 17, 114–125 (2014). https://doi.org/10.1016/j.nonrwa.2013.10.010Santonja, F., Shaikhet, L.: Analysing social epidemics by delayed stochastic models. Discret. Dyn. Nat. Soc. 2012, 13 (2012). https://doi.org/10.1155/2012/530472 . (Article ID 530472)Liu, L., Caraballo, T.: Analysis of a stochastic 2D-Navier-Stokes model with infinite delay. J. Dyn. Differ. Equ. pp 1–26 (2018). https://doi.org/10.1007/s10884-018-9703-xCaraballo, T., Colucci, R., Guerrini, L.: On a predator prey model with nonlinear harvesting and distributed delay. Commun. Pure Appl. Anal. 17(6), 2703–2727 (2018). https://doi.org/10.3934/cpaa.2018128Smith, R.C.: Uncertainty Quantification. Theory, Implementation and Applications. SIAM, Philadelphia (2014)Soong, T.T.: Random Differential Equations in Science and Engineering. Academic Press, New York (1973)Nouri, K., Ranjbar, H.: Mean square convergence of the numerical solution of random differential equations. Mediterr. J. Math. 12(3), 1123–1140 (2015). https://doi.org/10.1007/s00009-014-0452-8Zhou, T.: A stochastic collocation method for delay differential equations with random input. Adv. Appl. Math. Mech. 6(4), 403–418 (2014). https://doi.org/10.4208/aamm.2012.m38Shi, W., Zhang, C.: Generalized polynomial chaos for nonlinear random delay differential equations. Appl. Numer. Math. 115, 16–31 (2017). https://doi.org/10.1016/j.apnum.2016.12.004Lupulescu, V., Abbas, U.: Fuzzy delay differential equations. Fuzzy Optim. Decis. Mak. 11(1), 91–111 (2012). https://doi.org/10.1007/s10700-011-9112-7Liu, S., Debbouche, A., Wang, J.R.: Fuzzy delay differential equations. On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths. J. Comput. Appl. Math. 312, 47–57 (2017). https://doi.org/10.1016/j.cam.2015.10.028Krapivsky, P.L., Luck, J.L., Mallick, K.: On stochastic differential equations with random delay. J. Stat. Mech. Theory Exp. (2011). https://doi.org/10.1088/1742-5468/2011/10/P10008Garrido-Atienza, M.J., Ogrowsky, A., Schmalfuss, B.: Random differential equations with random delays. Stoch. Dyn. 11(2–3), 369–388 (2011). https://doi.org/10.1142/S0219493711003358Khusainov, D.Y., Ivanov, A.F., Kovarzh, I.V.: Solution of one heat equation with delay. Nonlinear Oscil. 12, 260–282 (2009). https://doi.org/10.1007/s11072-009-0075-3Asl, F.M., Ulsoy, A.G.: Analysis of a system of linear delay differential equations. J. Dyn. Syst. Meas. Control 125, 215–223 (2003). https://doi.org/10.1115/1.1568121Kyrychko, Y.N., Hogan, S.J.: On the use of delay equations in engineering applications. J. Vib. Control 16(7–8), 943–960 (2010). https://doi.org/10.1177/1077546309341100Villafuerte, L., Braumann, C.A., Cortés, J.C., Jódar, L.: Random differential operational calculus: theory and applications. Comput. Math. Appl. 59(1), 115–125 (2010). https://doi.org/10.1016/j.camwa.2009.08.061Strand, J.L.: Random ordinary differential equations. J. Diff. Equ. 7(3), 538–553 (1970). https://doi.org/10.1016/0022-0396(70)90100-2Khusainov, D.Y., Pokojovy, M.: Solving the linear 1d thermoelasticity equations with pure delay. Int. J. Math. Math. Sci. 2015, 1–11 (2015). https://doi.org/10.1155/2015/47926

Approximate solutions of randomized non-autonomous complete linear differential equations via probability density functions

[EN] Solving a random differential equation means to obtain an exact or approximate expression for the solution stochastic process, and to compute its statistical properties, mainly the mean and the variance functions. However, a major challenge is the computation of the probability density function of the solution. In this article we construct reliable approximations of the probability density function to the randomized non-autonomous complete linear differential equation by assuming that the diffusion coefficient and the source term are stochastic processes and the initial condition is a random variable. The key tools to construct these approximations are the random variable transformation technique and Karhunen-Loeve expansions. The study is divided into a large number of cases with a double aim: firstly, to extend the available results in the extant literature and, secondly, to embrace as many practical situations as possible. Finally, a wide variety of numerical experiments illustrate the potentiality of our findings.This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017-89664-P. The author Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2019). Approximate solutions of randomized non-autonomous complete linear differential equations via probability density functions. Electronic Journal of Differential Equations. 2019:1-40. http://hdl.handle.net/10251/139661S140201

Dispatcher3 – Machine learning for efficient flight planning - Approach and challenges for data-driven prototypes in air transport

Machine learning techniques to support decision making processes are in trend. These are particularly relevant in the context of flight management where large datasets of planned and realised operations are available. Current operations experience discrepancies between planned and executed flight plan, these might be due to external factors (e.g. weather, congestion) and might lead to sub-optimal decisions (e.g. recovering delay (burning extra fuel) when no holding is expected at arrival and therefore it was no needed). Dispatcher3 produces a set of machine learning models to support flight crew pre-departure, with estimations on expected holding at arrival, runway in use and fuel usage, and the airline’s duty manager on pre-tactical actions, with models trained with a larger look ahead time for ATFM and reactionary delay estimations. This paper describes the prototype architecture and approach of Dispatcher3 with particular focus on the challenges faced by this type of data-driven machine learning models in the field of air transport ranging: from technical aspects such as data leakage to operational requirements such as the consideration and estimation of uncertainty. These considerations should be relevant for projects which try to use machine learning in the field of aviation in general

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-

The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function

[EN] This paper deals with the damped pendulum random differential equation: (X) over double dot(t)+2 omega(0)xi(X) over dot(t) + omega X-2(0)(t) = Y(t), t is an element of [0, T], with initial conditions X(0) = X-0 and (X) over dot(0) = X-1. The forcing term Y(t) is a stochastic process and X-0 and X-1 are random variables in a common underlying complete probability space (Omega, F, P). The term X(t) is a stochastic process that solves the random differential equation in both the sample path and in the L-P senses. To understand the probabilistic behavior of X(t), we need its joint finite-dimensional distributions. We establish mild conditions under which X(t) is an absolutely continuous random variable, for each t, and we find its probability density function f(X(t))(x). Thus, we obtain the first finite-dimensional distributions. In practice, we deal with two types of forcing term: Y(t) is a Gaussian process, which occurs with the damped pendulum stochastic differential equation of Ito type; and Y(t) can be approximated by a sequence {Y-N(t)}(N-1)(infinity) in L-2([0, T] x Omega), which occurs with Karhunen-Loeve expansions and some random power series. Finally, we provide numerical examples in which we choose specific random variables X-0 and X-1 and a specific stochastic process Y(t), and then, we find the probability density function of X(t). (C) 2018 Elsevier B.V. All rights reserved.This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the valuable comments raised by the reviewers that have improved the final version of the paper.Calatayud-Gregori, J.; Cortés, J.; Jornet-Sanz, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A Statistical Mechanics and its Applications. 512:261-279. https://doi.org/10.1016/j.physa.2018.08.024S26127951