We apply Cartan’s method of equivalence to find a Bäcklund autotransformation for the tangent covering of the universal hierarchy equation. The transformation provides a recursion operator for symmetries of this equation.
We shall prove the following Theorem. Let Fs and Fu be two continuous transverse foliations with uniformly smooth leaves, of some manifold. If f is uniformly smooth along the leaves of Fs and Fu, then f is smooth.
We prove a formula relating the index of a solution and the rotation number of a certain complex vector along bifurcation diagrams.
It is known that the vector stop operator with a convex closed characteristic of class is locally Lipschitz continuous in the space of absolutely continuous functions if the unit outward normal mapping is Lipschitz continuous on the boundary of . We prove that in the regular case, this condition is also necessary.
Let be a field. The generalized Leibniz rule for higher derivations suggests the definition of a coalgebra for any positive integer . This is spanned over by , and has comultiplication and counit defined by and (Kronecker’s delta) for any . This note presents a representation of the coalgebra by using smooth spaces and a procedure of microlocalization. The author gives an interpretation of this result following the principles of the quantum theory of geometric spaces.