On the empirical eigenvalue distribution of slotted amplify-and-forward relaying protocol model

In this paper we consider the case of large cooperative communication systems where terminals use the protocol known as slotted amplify-and-forward protocol to aid the source in its transmission. Using the perturbation expansion methods of resolvents and large deviation techniques we obtain an expression for the Stieltjes transform of the asymptotic eigenvalue distribution of a sample covariance random matrix of the type HH† where H is the channel matrix of the transmission model for the transmission protocol we consider. We prove that the resulting expression is similar to the Stieltjes transform in its quadratic equation form for the Marcenko-Pastur distribution.

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy

An evaluation of alternative media for pebble matrix filtration using clay balls and recycled crushed glass

Protecting slow sand filters (SSFs) from high-turbidity waters by pretreatment using pebble matrix filtration (PMF) has previously been studied in the laboratory at University College London, followed by pilot field trials in Papua New Guinea and Serbia. The first full-scale PMF plant was completed at a water-treatment plant in Sri Lanka in 2008, and during its construction, problems were encountered in sourcing the required size of pebbles and sand as filter media. Because sourcing of uniform-sized pebbles may be problematic in many countries, the performance of alternative media has been investigated for the sustainability of the PMF system. Hand-formed clay balls made at a 100-yearold brick factory in the United Kingdom appear to have satisfied the role of pebbles, and a laboratory filter column was operated by using these clay balls together with recycled crushed glass as an alternative to sand media in the PMF. Results showed that in countries where uniform-sized pebbles are difficult to obtain, clay balls are an effective and feasible alternative to natural pebbles. Also, recycled crushed glass performed as well as or better than silica sand as an alternative fine media in the clarification process, although cleaning by drainage was more effective with sand media. In the tested filtration velocity range of ð0:72–1:33Þ m=h and inlet turbidity range of (78–589) NTU, both sand and glass produced above 95% removal efficiencies. The head loss development during clogging was about 30% higher in sand than in glass media.

Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations

Maximum-likelihood estimates of the parameters of stochastic differential equations are consistent and asymptotically efficient, but unfortunately difficult to obtain if a closed-form expression for the transitional probability density function of the process is not available. As a result, a large number of competing estimation procedures have been proposed. This article provides a critical evaluation of the various estimation techniques. Special attention is given to the ease of implementation and comparative performance of the procedures when estimating the parameters of the Cox–Ingersoll–Ross and Ornstein–Uhlenbeck equations respectively.