Abstract: In this paper, we investigate the low-lying energy
levels of the two-dimensional parabolic graphene quantum dots
(GQDs) in the presence of topological defects with long range
Coulomb impurity and subjected to an external uniform magnetic
field. The low-lying energy levels of the system are obtained within
the framework of the perturbation theory. We theoretically
demonstrate that a valley splitting can be controlled by geometrical
parameters of the graphene quantum dots and/or by tuning a uniform
magnetic field, as well as topological defects. It is found that, for
parabolic graphene dots, the valley splitting occurs due to the
introduction of spatial confinement. The corresponding splitting is
enhanced by the introduction of a uniform magnetic field and it
increases by increasing the angle of the cone in subcritical regime.
Abstract: The present study is concerned with the problem of determining the shape of the free surface flow in a hydraulic channel which has an uneven bottom. For the mathematical formulation of the problem, the fluid of the two-dimensional irrotational steady flow in water is assumed inviscid and incompressible. The solutions of the nonlinear problem are obtained by using the usual conformal mapping theory and Hilbert’s technique. An experimental study, for comparing the obtained results, has been conducted in a hydraulic channel (subcritical regime and supercritical regime).