Sextic anharmonic oscillators and orthogonal polynomials

Under certain constraints on the parameters a, b and c, it is known that Schroedinger's equation -y"(x)+(ax^6+bx^4+cx^2)y(x) = E y(x), a > 0, with the sextic anharmonic oscillator potential is exactly solvable. In this article we show that the exact wave function y is the generating function for a set of orthogonal polynomials P_n^{(t)}(x) in the energy variable E. Some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi-exact solvability. We also prove that this set of orthogonal polynomials can be reduced,by means of a simple scaling transformation, to a remarkable class of orthogonal polynomials, P_n(E)=P_n^{(0)}(E) recently discovered by Bender and Dunne.Comment: 11 page

Longitudinal Dynamic versus Kinematic Models for Car-Following Control Using Deep Reinforcement Learning

The majority of current studies on autonomous vehicle control via deep reinforcement learning (DRL) utilize point-mass kinematic models, neglecting vehicle dynamics which includes acceleration delay and acceleration command dynamics. The acceleration delay, which results from sensing and actuation delays, results in delayed execution of the control inputs. The acceleration command dynamics dictates that the actual vehicle acceleration does not rise up to the desired command acceleration instantaneously due to dynamics. In this work, we investigate the feasibility of applying DRL controllers trained using vehicle kinematic models to more realistic driving control with vehicle dynamics. We consider a particular longitudinal car-following control, i.e., Adaptive Cruise Control (ACC), problem solved via DRL using a point-mass kinematic model. When such a controller is applied to car following with vehicle dynamics, we observe significantly degraded car-following performance. Therefore, we redesign the DRL framework to accommodate the acceleration delay and acceleration command dynamics by adding the delayed control inputs and the actual vehicle acceleration to the reinforcement learning environment state, respectively. The training results show that the redesigned DRL controller results in near-optimal control performance of car following with vehicle dynamics considered when compared with dynamic programming solutions.Comment: Accepted to 2019 IEEE Intelligent Transportation Systems Conferenc

Wave equation and dispersion relations for a compressible rotating fluid

A fundamental non-classical fourth-order partial differential equation to describe small amplitude linear oscillations in a rotating compressible fluid, is obtained. The dispersion relations for such a fluid, and the different regions of the group and phase velocity are analyzed.Comment: 9 page