The purpose of this note is to state without proof certain. results on polynomial approximation to harmonic functions whose proofs will appear elsewhere later. In the plane of the complex variable z = x + iy, a (real) function u(z) harmonic in the interior of the Jordan curve C and continuous in the corresponding closed region C can be represented in C by a uniformly convergent series of harmon...

One may produce the qth harmonic of a string of length π by applying the ‘correct touch’ at the node π/q during a simultaneous pluck or bow. This notion was made precise by a model of Bamberger, Rauch and Taylor. Their ‘touch’ is a damper of magnitude b concentrated at π/q. The ‘correct touch’ is that b for which the modes, that do not vanish at π/q, are maximally damped. We here examine the as...

The Helmholtz equation arises in the study of electromagnetic radiation, optics, acoustics, etc. In spherical coordinates, its general solution can be written as a spherical harmonic series which satisfies the radiation condition at infinity, ensuring that the wave is outgoing. The boundary condition at infinity is hard to enforce with a finite element method since a suitable approximation need...

In this paper, separability of the perturbed 2-dimensional isotropic harmonic oscillators with homogeneous polynomial potentials is characterized from their Birkhoff-Gustavson (BG) normalization, one of the conventional methods for non-integrable Hamiltonian systems.

In this paper, we extend the method proposed by Cochelin and Vergez in [A high order purely frequency based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324:243–262, 2009] to the case of non polynomial non-linearities. This extension allows for the computation of branches of periodic solutions of a broader class of nonlinear dynamical syst...

The spectra of, e.g. open quantum systems are typically given as the superposition of resonances with a Lorentzian line shape, where each resonance is related to a simple pole in the complex energy domain. However, at exceptional points two or more resonances are degenerate and the resulting non-Lorentzian line shapes are related to higher order poles in the complex energy domain. In the Fourie...

We consider the one-dimensional quantum harmonic oscillator perturbed by a linear operator which is polynomial of degree $2$ in $(x,-{\rm i}\partial_x)$, with coefficients quasi-periodically depending on time. By establishing reducibility results, we describe growth Sobolev norms. In particular, $t^{2s}-$polynomial ${\mathcal H}^s-$norm observed this model if original time quasi-periodic equati...

It is well known that the real cohomology of a compact Riemannian manifold M is isomorphic to the algebra of its harmonic forms. When M is a fiat Riemannian manifold, i.e. a Euclidean manifold, a differential form is harmonic if and only if it is a parallel differential form. In the local Euclidean coordinate systems on M, such a differential form has constant coefficients. Thus, for a Euclidea...