Axioms, Volume 3, Issue 3 (September 2014) – 3 articles , Pages 300-341

This paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by others and the space derivative of second order by the Riesz–Feller fractional derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space fractional derivatives, are also investigated. Full article

A fundamental tool in shape analysis is the virtual embedding of the Riemannian manifold describing the geometry of a shape into Euclidean space. Several methods have been proposed to embed isometric shapes into flat domains, while preserving the distances measured on the manifold. Recently, attention has been given to embedding shapes into the eigenspace of the Laplace–Beltrami operator. The Laplace–Beltrami eigenspace preserves the diffusion distance and is invariant under isometric transformations. However, Laplace–Beltrami eigenfunctions computed independently for different shapes are often incompatible with each other. Applications involving multiple shapes, such as pointwise correspondence, would greatly benefit if their respective eigenfunctions were somehow matched. Here, we introduce a statistical approach for matching eigenfunctions. We consider the values of the eigenfunctions over the manifold as the sampling of random variables and try to match their multivariate distributions. Comparing distributions is done indirectly, using high order statistics. We show that the permutation and sign ambiguities of low order eigenfunctions can be inferred by minimizing the difference of their third order moments. The sign ambiguities of antisymmetric eigenfunctions can be resolved by exploiting isometric invariant relations between the gradients of the eigenfunctions and the surface normal. We present experiments demonstrating the success of the proposed method applied to feature point correspondence. Full article