Multiresolutional S-search generated a need to properly tessellate spaces and efficiently searching them. Drexel University has introduced MR-methodology such as uniform and non-uniform space tessellation and efficient algorithms of searching within ressellated state space. This methodology for solving planning and control problems is successfully applied in autonomous vehicles, industrial robo...

We introduce tessellation of the filled Julia sets for hyperbolic and parabolic quadratic maps. Then the dynamics inside their Julia sets are organized by tiles which work like external rays outside. We also construct continuous families of pinching semiconjugacies associated with hyperblic-to-parabolic degenerations without using quasiconformal deformation. Instead we use tessellation and inve...

A tessellation is a process that surrounds us every day. It is an old problem in mathematics and in computational geometry Preparata, 1985 . Several tessellations are known, although only some of them are interesting in the computational geometry. The main aim of the tessellation is to tile or divide the space of the interest into smaller parts tiles upon which other problems could be solved mo...

A centroidal Voronoi tessellation is a Voronoi tessellation whose generating points are the centroids (centers of mass) of the corresponding Voronoi regions. We give some applications of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals. We discuss methods f...

There are various models of first passage percolation (FPP) in R. We want to start a very general study of this topic. To this end we generalize the first passage percolation model on the lattice Z to R and adapt the results of [Boi90] to prove a shape theorem for ergodic random pseudometrics on R. A natural application of this result will be the study of FPP on random tessellations where a flu...