Necessary and Sufficient Conditions
The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal algebraically generated by the collection of αi inside the ring Ω(M) is differentially closed, in other words
then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.)
Read more about this topic: Integrability Conditions For Differential Systems
Famous quotes containing the words sufficient conditions, sufficient and/or conditions:
“We as a nation need to be reeducated about the necessary and sufficient conditions for making human beings human. We need to be reeducated not as parents—but as workers, neighbors, and friends; and as members of the organizations, committees, boards—and, especially, the informal networks that control our social institutions and thereby determine the conditions of life for our families and their children.”
—Urie Bronfenbrenner (b. 1917)
“They who do not understand that a man may be brought to hope that which of all things is the most grievous to him, have not observed with sufficient closeness the perversity of the human mind.”
—Anthony Trollope (1815–1882)
“There is no society known where a more or less developed criminality is not found under different forms. No people exists whose morality is not daily infringed upon. We must therefore call crime necessary and declare that it cannot be non-existent, that the fundamental conditions of social organization, as they are understood, logically imply it.”
—Emile Durkheim (1858–1917)