We consider a static, spherically symmetric system of a Dirac particle in a classical gravitational and SU (2) Yang-Mills field. We prove that the only black-hole solutions of the corresponding Einstein-Dirac-Yang/Mills equations are the Bartnik-McKinnon black-hole solutions of the SU (2) Einstein-Yang/Mills equations; thus the spinors must vanish identically. This indicates that the Dirac particles must either disappear into the black-hole or escape to infinity.

The Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole, thereby putting supperradiance, the wave analogue of the Penrose process, into a rigorous mathematical framework. We quantify the maximal energy gain. We also compute the infinitesimal change of mass and angular momentum of the black hole, in agreement with Christodoulous result for the Penrose process. The main mathematical tool is our previously derived integral representation of the wave propagator.

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In this paper we construct a new time-periodic solution of the vacuum Einsteins field equations, this solution possesses physical singularities, i.e., the norm of the solutions Riemann curvature tensor takes the infinity at some points. We show that this solution is intrinsically time-periodic and describes a time-periodic universe with the time-periodic physical singularity. By calculating the Weyl scalars of this solution, we investigate new physical phenomena and analyze new singularities for this universal model.

In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of graph Hamiltonians, being generic in a sense, inspires the question about the existence of graphs with a finite and nonzero number of spectral gaps. We show that the answer depends on the vertex couplings together with commensurability of the graph edges. A finite and nonzero number of gaps is excluded for graphs with scale invariant couplings; on the other hand, we demonstrate that graphs featuring a finite nonzero number of gaps do exist, illustrating the claim on the example of a rectangular lattice with a suitably tuned -coupling at the vertices.