Modelling acoustics on the Poincaré half-plane

Modelling acoustics on the Poincaré half-plane. Michael M. Tung. Journal of Computational and Applied Mathematics DOI10.1016/j.cam.2017.10.037

Abstract: Novel advances in the field of metamaterial research have permitted the engineering of devices with extraordinary characteristics. Here, we explore the possibilities in transformation acoustics to implement a model for the simulation of acoustic wave propagation on the Poincaré half-plane-the simplest model possessing hyperbolic geometry and also of considerable historical interest. We start off from a variational principle on the given spacetime manifold to find the design description of the model in the laboratory. After examining some significant geometrical and physical properties of the Poincaré half-plane model, we derive a general formal solution for its acoustic wave propagation. A numerical example for the evolution of the acoustic potential on a rectangular region of the Poincaré half-plane concludes this discussion.

A new type of Hermite matrix polynomial series

A new type of Hermite matrix polynomial series (E. DEFEZ, M. TUNG), Quaestiones Mathematicae, 41(2), 2018, 205–212.

Abstract: Conventional Hermite polynomials emerge in a great diversity of applications
in mathematical physics, engineering, and related fields. However, in physical
systems with higher degrees of freedom it will be of practical interest to extend
the scalar Hermite functions to their matrix analogue. This work introduces various
new generating functions for Hermite matrix polynomials and examines existence and
convergence of their associated series expansion by using Mehlerís formula for the
general matrix case. Moreover, we derive interesting new relations for even- and
odd-power summation in the generating-function expansion containing Hermite matrix
polynomials. Some new results for the scalar case are also presented.