A proof of the Riemann’s hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D in one dimension, and their respective eigenfunctions ψs(t), parameterized by continuous real indexes k and l. Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function. Due to the fu...

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [19], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting...

We describe a computational method to compute spectra and slowly decaying eigenfunctions of linearizations of the cubic–quintic complex Ginzburg– Landau equation about numerically determined stationary solutions. We compare the results of the method to a formula for an edge bifurcation obtained using the small dissipation perturbation theory of Kapitula and Sandstede. This comparison highlights...

The inverse Sturm-Liouville problem finds its applications in the identification of mechanical properties and/or geometrical configurations of a vibrating continuous medium; however, this problem is hard to solve, either theoretically or numerically. Previously, Liu (2008a) has constructed a Lie-group shooting method to determine the eigenvalues, and the corresponding eigenfunctions, for the di...