Equilibrium distributions of discrete non-autonomous graphs

We introduce the notions of equilibrium distribution and time of convergence in discrete non-autonomous graphs. Under some conditions we give an estimate to the convergence time to the equilibrium distribution using the second largest eigenvalue of some matrices associated with the system.

Periodic paths on nonautonomous graphs

We define nonautonomous graphs as a class of dynamic graphs in discrete time whose time-dependence consists in connecting or disconnecting edges. We study periodic paths in these graphs, and the associated zeta functions. Based on the analytic properties of these zeta functions we obtain explicit formulae for the number of n-periodic paths, as the sum of the nth powers of some specific algebraic numbers.

Behaviour and fate of metals in urban wastewater treatment plants: a review

The concerns on metals in urban wastewater treatment plants (WWTPs) are mainly related to its contents in discharges to environment, namely in the final effluent and in the sludge produced. In the near future, more restrictive limits will be imposed to final effluents, due to the recent guidelines of the European Water Framework Directive (EUWFD). Concerning the sludge, at least seven metals (Cd, Cr, Cu, Hg, Ni, Pb and Zn) have been regulated in different countries, four of which were classified by EUWFD as priority substances and two of which were also classified as hazardous substances. Although WWTPs are not designed to remove metals, the study of metals behaviour in these systems is a crucial issue to develop predictive models that can help more effectively the regulation of pre-treatment requirements and contribute to optimize the systems to get more acceptable metal concentrations in its discharges. Relevant data have been published in the literature in recent decades concerning the occurrence/fate/behaviour of metals in WWTPs. However, the information is dispersed and not standardized in terms of parameters for comparing results. This work provides a critical review on this issue through a careful systematization, in tables and graphs, of the results reported in the literature, which allows its comparison and so its analysis, in order to conclude about the state of the art in this field. A summary of the main consensus, divergences and constraints found, as well as some recommendations, is presented as conclusions, aiming to contribute to a more concerted action of future research. © 2015, Islamic Azad University (IAU).

Spectral and dynamical invariants in a complete clustered network

The main result of this work is a new criterion for the formation of good clusters in a graph. This criterion uses a new dynamical invariant, the performance of a clustering, that characterizes the quality of the formation of clusters. We prove that the growth of the dynamical invariant, the network topological entropy, has the effect of worsening the quality of a clustering, in a process of cluster formation by the successive removal of edges. Several examples of clustering on the same network are presented to compare the behavior of other parameters such as network topological entropy, conductance, coefficient of clustering and performance of a clustering with the number of edges in a process of clustering by successive removal.

Nonautonomous Graphs and Topological Entropy of Nonautonomous Lorenz Systems

In this work, we associate a p-periodic nonautonomous graph to each p-periodic nonautonomous Lorenz system with finite critical orbits. We develop Perron-Frobenius theory for nonautonomous graphs and use it to calculate their entropy. Finally, we prove that the topological entropy of a p-periodic nonautonomous Lorenz system is equal to the entropy of its associated nonautonomous graph.