Modern cryptography is largely based on complexity assumptions, for example, the ubiquitous RSA is based on the supposed complexity of the prime factorization problem. Thus, it is of fundamental importance to understand how a quantum computer would e
ventually weaken these algorithms. In this paper, one follows Feynmans prescription for a computer to simulate the physics corresponding to the algorithm of factoring a large number $N$ into primes. Using Dirac-Jordan transformation theory one translates factorization into the language of quantum hermitical operators, acting on the vectors of the Hilbert space. This leads to obtaining the ensemble of factorization of $N$ in terms of the Euler function $varphi(N)$, that is quantized. On the other hand, considering $N$ as a parameter of the computer, a Quantum Mechanical Prime Counting Function $pi_{QM}(x)$, where $x$ factorizes $N$, is derived. This function converges to $pi(x)$ when $Ngg x$. It has no counterpart in analytic number theory and its derivation relies on semiclassical quantization alone.
In this work we study thermoelectric properties of graphene nanoribbons with side-attached organic molecules. By adopting a single-band tight binding Hamiltonian and the Greens function formalism, we calculated the transmission and Seebeck coefficien
ts for different hybrid systems. The corresponding thermopower profiles exhibit a series of sharp peaks at the eigenenergies of the isolated molecule. We study the effects of the temperature on the thermoelectric response, and we consider random configurations of molecule distributions, in different disorder regimes. The main characteristics of the thermopower are not destroyed under temperature and disorder, indicating the robustness of the system as a proposed molecular thermo-sensor device.
We show that a bilayer graphene flake deposited above a ferromagnetic insulator can behave as a spin-filtering device. The ferromagnetic material induces exchange splitting in the graphene flake, and due to the Fano antiresonances occurring in the tr
ansmission of the graphene flake as a function of flake length and energy, it is possible to obtain a net spin current. This happens when an antiresonance for one spin channel coincides with a maximum transmission for the opposite spin. We propose these structures as a means to obtain spin-polarized currents and spin filters in graphene-based systems.
We apply the postquasistatic approximation, an iterative method for the evolution of self-gravitating spheres of matter, to study the evolution of dissipative and electrically charged distributions in General Relativity. We evolve nonadiabatic distri
butions assuming an equation of state that accounts for the anisotropy induced by the electric charge. Dissipation is described by streaming out or diffusion approximations. We match the interior solution, in noncomoving coordinates, with the Vaidya-Reissner-Nordstrom exterior solution. Two models are considered: i) a Schwarzschild-like shell in the diffusion limit; ii) a Schwarzschild-like interior in the free streaming limit. These toy models tell us something about the nature of the dissipative and electrically charged collapse. Diffusion stabilizes the gravitational collapse producing a spherical shell whose contraction is halted in a short characteristic hydrodynamic time. The streaming out radiation provides a more efficient mechanism for emission of energy, redistributing the electric charge on the whole sphere, while the distribution collapses indefinitely with a longer hydrodynamic time scale.
