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 words ring of and/or ring:

    “The Mormons make the marriage ring, like the ring of Saturn, fluid, not solid, and keep it in its place by numerous satellites.”
    —Henry Wadsworth Longfellow (1807–1882)

    “Close friends contribute to our personal growth. They also contribute to our personal pleasure, making the music sound sweeter, the wine taste richer, the laughter ring louder because they are there.”
    —Judith Viorst (20th century)