Splitting the one-Dimensional Wave Equation, Part II: Additional Data are Given by an End Displacement Measurement

In this research, an unknown space-dependent force function in the wave equation is studied. This is a natural continuation of [1] and chapter 2 of [2] and [3], where the finite difference method (FDM)/boundary element method (BEM), with the separation of variables method, were considered. Additional data are given by the one end displacement measurement. Moreover, it is a continuation of [3], with exchanging the boundary condition, where are extra data, by the initial condition. This is an ill-posed inverse force problem for linear hyperbolic equation. Therefore, in order to stabilize the solution, a zeroth-order Tikhonov regularization method is provided. To assess the accuracy, the minimum error between exact and numerical solutions for the force is computed for various regularization parameters. Numerical results are presented and a good agreement was obtained for the exact and noisy data.

Black-hole dynamics

The main ideas from black-hole perturbation theory are introduced, starting with stability isses and leading on to the notion of quasinormal modes. The motion of test bodies is considered, making it possible to estimate the gravitational waves emitted in a black-hole merger, and issues associated with the self-force problem are considered.

On the n -body problem in R4

Using geometric mechanics methods, we examine aspects of the dynamics of n mass points in R 4 with a general pairwise potential. We investigate the central force problem, set up the n -body problem and discuss certain properties of relative equilibria. We describe regular n -gons in R 4 , and when the masses are equal we determine the invariant manifold of motions with regular n -gon configurations. In the case n = 3, we reduce the dynamics to a 6 d.f. system and we show that for generic potentials and momenta, relative equilibria with equilateral configuration are unstable. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

No Need for Dark-Matter, Dark-Energy or Inflation, Once Ordinary Matter Is Properly Represented?

In a recent Foundations of Physics paper [5] by the current author it was shown that, when the self-force problem of classical electrodynamics is properly solved, it becomes a plausible ontology underlying the statistical description of quantum mechanics. In the current paper we extend this result, showing that ordinary matter, thus represented, possibly suffices in explaining the outstanding observations currently requiring for this task the contrived notions of dark-matter, dark-energy and inflation. The single mandatory 'fix' to classical electrodynamics, demystifying both very small and very large scale physics, should be contrasted with other adhoc solutions to either problems. Instrumental to our cosmological model is scale covariance (and 'spontaneous breaking' thereof), a formal symmetry of classical electrodynamics treated on equal footing with its Poincare covariance, which is incompatible with the (absolute) metrical attributes of the GR metric tensor.