Submissions from F. F. Serafini
 [1] faKiv:2102.07600 [pdf]

The empty sphere and the concept of dilemmasComments: 23 pages, title changed, minor modifications
The discussion of the empty sphere is accompanied by the definition of a novel structure for the Euclidean empty sphere, which is the quasipolygonal form of the geometry of a quiver of the geometrical string. The structure of the empty sphere is extended to include a nonpolygonal form. The definition of a nonpolygonal form of the empty sphere is based on the existence of the nonpolygonal form in the Geometric Formalism. The geometry of the empty sphere is defined for a given vector bundle, which is the inverse of the Euclidean geometrical unit bundle. The definition of the empty sphere is derived by using the symmetric geometrical form of the geometrical string.
 [2] faKiv:2102.07601 [pdf]

TransPlanckian wave equations in the Minkowski stateComments: 23 pages, 2 figures
We analyze the transPlanckian wave equations in Minkowski state at the Minkowski point, arising from the presence of the nonpolygonal wave equation in the Minkowski state.