Narrow Results

Electromagnetic waves, solving the full set of Maxwell equations in vacuum, are numerically computed. These waves occupy a fixed bounded region of the three-dimensional space, topologically equivalent to a toroid. Thus, their fluid dynamics analogs are vortex rings. An analysis of the shape of the sections of the rings, depending on the angular speed of rotation and the major diameter, is carried out. Successively, spherical electromagnetic vortex rings of Hill's type are taken into consideration. For some interesting peculiar configurations, explicit numerical solutions are exhibited.

Computing the Hausdorff measure of C × C, where C is the classical ternary Cantor set, is a long standing difficult problem. It is well-known that for a self-similar set, calculating the Hausdorff measure is equivalent to determining its optimal sets.

This paper studies optimal sets of C × C: their diameters, measures, symmetries and the shapes. For this purpose, we introduce several devices: the repulsive principle, a bipartite graph G and a W-function. We show that the diameter of the optimal set B is between 1.2993 and 1.3082. Two symmetry properties of B are proved. Finally, we show that the shape of B is very close to a disk. We conjecture that an optimal set might be a disk.