Differential Algebra - Ring of Pseudo-differential Operators

Ring of Pseudo-differential Operators

Differential rings and differential algebras are often studied by means of the ring of pseudo-differential operators on them.

This is the ring

Multiplication on this ring is defined as

(r\xi^m)(s\xi^n) = \sum_{k=0}^m r (\partial^k s) {m \choose k} \xi^{m+n-k}.

Here is the binomial coefficient. Note the identities

which makes use of the identity

and

Read more about this topic:  Differential Algebra

Famous quotes containing the word ring:

    “I started out very quiet and I beat Mr. Turgenev. Then I trained hard and I beat Mr. de Maupassant. I’ve fought two draws with Mr. Stendhal, and I think I had an edge in the last one. But nobody’s going to get me in any ring with Mr. Tolstoy unless I’m crazy or I keep getting better.”
    —Ernest Hemingway (1899–1961)