Approximation properties of the qq-sine bases

For $q>12/11$ the eigenfunctions of the non-linear eigenvalue problem associated to the one-dimensional $q$-Laplacian are known to form a Riesz basis of $L^2(0,1)$. We examine in this paper the approximation properties of this family of functions and its dual, in order to establish non-orthogonal spectral methods for the $p$-Poisson boundary value problem and its corresponding parabolic time evolution initial value problem. The principal objective of our analysis is the determination of optimal values of $q$ for which the best approximation is achieved for a given $p$ problem.Comment: 20 pages, 11 figures and 2 tables. We have fixed a number of typos and added references. Changed the title to better reflect the conten

Equation-free analysis of a dynamically evolving multigraph

In order to illustrate the adaptation of traditional continuum numerical techniques to the study of complex network systems, we use the equation-free framework to analyze a dynamically evolving multigraph. This approach is based on coupling short intervals of direct dynamic network simulation with appropriately-defined lifting and restriction operators, mapping the detailed network description to suitable macroscopic (coarse-grained) variables and back. This enables the acceleration of direct simulations through Coarse Projective Integration (CPI), as well as the identification of coarse stationary states via a Newton-GMRES method. We also demonstrate the use of data-mining, both linear (principal component analysis, PCA) and nonlinear (diffusion maps, DMAPS) to determine good macroscopic variables (observables) through which one can coarse-grain the model. These results suggest methods for decreasing simulation times of dynamic real-world systems such as epidemiological network models. Additionally, the data-mining techniques could be applied to a diverse class of problems to search for a succint, low-dimensional description of the system in a small number of variables