We present here a theory of the electronic properties of quasi two-dimensional quantum dots made of topological insulators. The topological insulator is described by either eight band k→·p→ Hamiltonian or by a four-band k→·p→ Bernevig–Hughes–Zhang (BHZ) Hamiltonian. The trivial versus topological properties of the BHZ Hamiltonian are characterized by the different topologies that arise when mapping the in-plane wavevectors through the BHZ Hamiltonian onto a Bloch sphere. In the topologically nontrivial case, edge states are formed in the disc and square geometries of the quantum dot. We account for the effects of compressive strain in topological insulator quantum dots by means of the Bir–Pikus Hamiltonian. Tuning strain allows topological phase transitions between topological and trivial phases, which results in the vanishing of edge states from the energy gap. This may enable the design of a quantum strain sensor based on strain-driven transitions in HgTe topological insulator square quantum dots.
A black hole with zero angular momentum is said to be stationary and under certain conditions such a black hole can represented as a sphere. This review examines Hawking’s topology theorem, the Schwarzschild metric, novel solutions to Einstein’s equations, resonances of hyperbolic orbits around the event horizon for spherical, stationary black holes, and analyzes their importance. It is suggested, that in the spherical stationary black hole case, the Fourier analysis can be used to find the resonances due to Geometric scattering of hyperbolic orbits and thus the outgoing energy fields from the event horizon can be found more precisely; allowing for the adequate signal processing analysis to be found for such a field.
The Kerr black hole rotates with two parameters: mass M and angular momentum a and is characterized by the Kerr metric (Taylor and Wheeler 2000). Hence, a binary pair of a black hole and a star can create an accretion disc. A Kerr ray tracer algorithm was used to simulate accretion discs in the Seyfert-1 galaxy. The power law observed flux of relativistic emission lines, and Kerr Fourier image analysis methods were applied to the simulated discs. Simulated image characteristics were analyzed. Power laws were fitted to the simulated data of the Mrk110 accretion disc. Lastly, the simulated images were transformed into Fourier space and characteristics were discussed.
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