1. Numerical Comparison Between RBF Schemes With Respect to Other Approaches to Solve Fractional Partial Differential Equations and Their Advantages When Choosing Non-Uniform Nodes

    Carlos Alberto Torres Martinez1,, Fernando Brambila Paz2

    1. PhD student at Facultad de Ciencias UNAM and professor at Universidad Autonoma de la Ciudad de Mexico (UACM).
    2. Proffesor and researcher at Facultad de Ciencias UNAM, Mexico.

    Abstract: In this work, some cases of Fractional Partial Differential Equations (FPDE) are considered and resolved numerically using a meshless method via Radial Basis Functions (RBF). Taking different types of fractional derivatives and nonuniform collocation nodes. The results are compared with those obtained by previous works. This type of approach sounds like a good choice to deal with problems in higher dimensions or non-uniform data.

    Keywords: Fractional Partial Differential Equation (FPDE), Meshless Methods, Radial Basis Functions (RBF), fractional diffusion-wave equation (DWFE), Numerical solution.

    Pages: 85 – 105 | Full PDF Paper
  2. Combining MM-Algorithms and MCMC Procedures for Maximum Likelihood Estimation in Mallows-Bradley-Terry Models

    Amadou Sawadogo1, Dominique Lafon2, Simplice Dossou-Gbété3

    1. Department of Mathematics and Computer, University of Félix Houphouet-Boigny, Ivory Coast, Abidjan.
    2. Ecole des Mines d’ Alès, Site Hélioparc, Pau, France.
    3.University of Pau et des Pays de l’Adour/CNRS. Laboratory of Mathematics and their Application of Pau-IPRA, UMR CNRS 5142. Pau, France.

    Abstract: This paper is devoted to the computation of the maximum likelihood estimates of the Mallows-Bradley-Terry ranking model parameters. The maximum likelihood method is avoid because of the normalizing constant that may involve an untractable sum with a very large number of terms. We show how to implement a Monte Carlo Maximization-Minimization algorithm to estimate the model parameters: the evaluation of the mathematical expectations involved in the log-likelihood equation is obtained by generating samples of Monte Carlo Markov chain from the stationary distribution. In addition, a simulation study for asymptotic properties assessment has been made. The proposed method is applied to analyze real life data set of the literature. The present paper is restricted to the Mallows-Bradley-Terry ranking model that does not allow for possibility of ties. This case has been studied elsewhere.

    Keywords: Mallows-Bradley-Terry model, rank data, maximum likelihood method, MM-algorithm, Gibbs sampling.

    Pages: 106 – 128 | Full PDF Paper