We study the regularity properties of inverse a bilipschitz mapping f belonging to W m X loc $W^m X_{\mathrm{loc}}$ , where is an arbitrary Banach function space. Namely, we prove that − 1 $f^{-1}$ also in . Furthermore, paper shows class mappings closed with respect composition and multiplication.

the mathematical aspect of cartographic mapping is a process which establishes a unique connection between points of the earth’s sphere and their images on a plane. it was proven in differential geometry that an isometric mapping of a sphere onto a plane with all corresponding distances on both surfaces remaining identical can never be achieved since the two surfaces do not possess the same gau...

To overcome shortcomings when the ant colony optimization clustering algorithm (ACOC) deal with problem, this paper introduces a novel chaos. The main idea of is to apply chaotic mapping function in two stages optimization: pheromone initialization and update. application phase can encourage ants be distributed as many different initial states possible. Applying update stage add disturbance fac...