A bstract The exceptional symmetry is realized perturbatively in F-theory which the manifest U-duality theory. SO(5) acts on both 16 space-time coordinates and 10 worldvolume coordinates. Closure of Virasoro algebra requires Gauss law constraints worldvolume. This set current algebras describes a 10-brane. duality enlarged to SO(6) Lagrangian formulation. We propose actions 10-brane with symmet...

This paper gives new upper bounds on the stick numbers of knots [Formula: see text], and as well equilateral number text]. These imply that text] have superbridge index completing project showing no prime through 10 crossings can larger than 5. The current best for are given in Appendix A.