Importance Sampling for Objetive Funtion Estimations in Neural Detector Traing Driven by Genetic Algorithms

To train Neural Networks (NNs) in a supervised way, estimations of an objective function must be carried out. The value of this function decreases as the training progresses and so, the number of test observations necessary for an accurate estimation has to be increased. Consequently, the training computational cost is unaffordable for very low objective function value estimations, and the use of Importance Sampling (IS) techniques becomes convenient. The study of three different objective functions is considered, which implies the proposal of estimators of the objective function using IS techniques: the Mean-Square error, the Cross Entropy error and the Misclassification error criteria. The values of these functions are estimated by IS techniques, and the results are used to train NNs by the application of Genetic Algorithms. Results for a binary detection in Gaussian noise are provided. These results show the evolution of the parameters during the training and the performances of the proposed detectors in terms of error probability and Receiver Operating Characteristics curves. At the end of the study, the obtained results justify the convenience of using IS in the training

On Aharonov-Casher bound states

In this work bound states for the Aharonov-Casher problem are considered. According to Hagen's work on the exact equivalence between spin-1/2 Aharonov-Bohm and Aharonov-Casher effects, is known that the $\boldsymbol{\nabla}\cdot\mathbf{E}$ term cannot be neglected in the Hamiltonian if the spin of particle is considered. This term leads to the existence of a singular potential at the origin. By modeling the problem by boundary conditions at the origin which arises by the self-adjoint extension of the Hamiltonian, we derive for the first time an expression for the bound state energy of the Aharonov-Casher problem. As an application, we consider the Aharonov-Casher plus a two-dimensional harmonic oscillator. We derive the expression for the harmonic oscillator energies and compare it with the expression obtained in the case without singularity. At the end, an approach for determination of the self-adjoint extension parameter is given. In our approach, the parameter is obtained essentially in terms of physics of the problem.Comment: 11 pages, matches published versio

THE AHARONOV CASHER SCATTERING: THE EFFECT OF THE del center dot E TERM

In the AharonovCasher (AC) scattering, a neutral particle interacts with an infinitesimally thin, long charge filament resulting in a phase shift. In the original AC treatment, del center dot E term proportional to the charge density at the filaments position is dropped from the Hamiltonian on the basis that the particle is banned from the filament, thus, the resulting Hamiltonian compares with the AharonovBohm Hamiltonian of a scalar particle. Here, we consider AC scattering with this term included. Starting from the three-dimensional nonrelativistic AharonovCasher (AC) Schrodinger equation with the del center dot E term included, we find the wave functions - in particular their singular component - the phase shifts and thus compute the scattering amplitudes and cross-sections. We show that singular solutions in the AC case appear only when the delta function interaction introduced is attractive regardless of the spin orientation of the particle. We find that the inclusion of this term does not introduce a structural difference in the general form of the cross-section even for polarized particles. Its mere effect, is in shifting the parameter N (the greatest integer in alpha) that appears in the cross-section, in the attractive case, by one. Interesting situation appears when N = 0, thus alpha = delta, in the case alpha > 0, and N = -1, so alpha - 1 - delta in the case alpha > 0: At these values of the parameter N, where alpha is just any fraction, the cross-section for a particle polarized in the scattering plane to scatter in a state with the same polarization, is isotropic. It also vanishes, at these values of N, for transitions between same-helicity eigenstates. For these values of the parameter N and at the special values alpha = +/- 1/2, the cross-sections for both signs of alpha coincide. The main differences between this model and the mathematically equivalent spin-1/2 AB theory are outlined