Localization properties of a tight-binding electronic model on the Apollonian network

An investigation on the properties of electronic states of a tight-binding Hamiltonian on the Apollonian network is presented. This structure, which is defined based on the Apollonian packing problem, has been explored both as a complex network, and as a substrate, on the top of which physical models can defined. The Schrodinger equation of the model, which includes only nearest neighbor interactions, is written in a matrix formulation. In the uniform case, the resulting Hamiltonian is proportional to the adjacency matrix of the Apollonian network. The characterization of the electronic eigenstates is based on the properties of the spectrum, which is characterized by a very large degeneracy. The $2\pi /3$ rotation symmetry of the network and large number of equivalent sites are reflected in all eigenstates, which are classified according to their parity. Extended and localized states are identified by evaluating the participation rate. Results for other two non-uniform models on the Apollonian network are also presented. In one case, interaction is considered to be dependent of the node degree, while in the other one, random on-site energies are considered.Comment: 7pages, 7 figure

Procedure to evaluate multivariate statistical process control using ARIMA-ARCH models

Technological development and production processes require statistical process control in the use of alternative techniques to evaluate a productive process. This paper proposes an alternative procedure for monitoring a multivariate productive process using residuals obtained from the principal component scores modeled by the general class of autoregressive integrated moving average (ARIMA) and the generalized autoregressive conditional heteroskedasticity (GARCH) processes. We seek to obtain and investigate non-correlated and independent residuals by means of X-bar and exponentially weighted moving average (EWMA) charts as a way to capture large and small variations in the productive process. The principal component analysis deals with the correlation among the variables and reduces the dimensions. The ARIMA-GARCH model estimates the mean and volatility of the principal components selected, providing independent residuals that are analyzed using control charts. Thus, a multivariate process can be assessed using univariate techniques, taking into account both the mean and the volatility behavior of the process. Therefore, we present an alternative procedure to evaluate a process with multivariate features to determine the level of volatility persistence in the productive process when an external action occurs