Some recent results on the zeros of Bessel functions and orthogonal polynomials

AbstractSeveral recent results are encountered on the zeros of Bessel functions and of the classical orthogonal polynomials, many of them are waiting for being published. Four problems are formulated as conjectures

Fast sampling from β\beta-ensembles

We study sampling algorithms for $\beta$-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of $\beta$-ensembles, and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size $N$ mixes in $\mathcal{O}(\log(N))$.Comment: 37 pages, 8 figures, code at https://github.com/guilgautier/DPP

Design of a model predictive controller to control UAVs

In this report a continuous time model predictive controller is applied to the control of an Unmanned Aerial Vehicle (UAV). The purpose of this work, is to see if it is possible to design a good tracking controller that is capable of simulating a highly nonlinear aircraft system. For the simulations a MATLAB Simulink model of the UAV 'Ariel ' is used. The nonlinear Simulink model covers all the aircraft dynamics. The model consists of five inputs, twelve states and twelve outputs. This model is linearized at several different operating conditions, for use with the model predictive controller. The model predictive controller is capable of simulating in real time, making it possible to use this in on-line applications. The Ariel model is linearized to a linear time invariant state space system. The state space model is then extended. This means the outputs of the original model are included in the state vector of the extended model. With this extended state space model, it can be shown that the selected outputs can be tracked by the controller. The controller uses an algorith